#347038

order	512
nilpotency class	3
isoclinism	isoclinic class of #347034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2070 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 2070 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(2,10)(3,16)(4,21)(5,23)(6,22)(11,38)(12,31)(13,46)(14,26)(15,37)(17,27)(18,52)(19,32)(20,50)(24,25)(28,33)(29,30)(34,48)(35,59)(36,41)(39,43)(40,47)(42,62)(44,55)(45,58)(49,51)(53,56)(54,61)(57,63)(60,64);(1,2)(3,11)(4,12)(5,26)(6,31)(7,34)(8,14)(9,15)(10,23)(13,22)(16,47)(17,39)(18,54)(19,43)(20,42)(21,55)(24,45)(25,46)(27,57)(28,58)(29,37)(30,48)(32,41)(33,44)(35,60)(36,56)(38,52)(40,51)(49,62)(50,61)(53,64)(59,63);(1,3)(2,11)(4,17)(5,27)(6,32)(7,35)(8,20)(9,19)(10,36)(12,39)(13,47)(14,42)(15,43)(16,22)(18,30)(21,49)(23,56)(24,53)(25,52)(26,57)(28,59)(29,51)(31,41)(33,50)(34,60)(37,40)(38,46)(44,61)(45,64)(48,54)(55,62)(58,63);(1,4,8,24)(2,12,14,45)(3,17,20,53)(5,28,23,6)(7,21,9,25)(10,31,26,58)(11,39,42,64)(13,48,44,37)(15,46,34,55)(16,18,50,51)(19,52,35,49)(22,30,33,29)(27,59,56,32)(36,41,57,63)(38,60,62,43)(40,47,54,61)

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72272601
maximal subgroup of	#72272561, #72272601
maximal normal subgroup of	#72272561, #72272601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254234);