#347034

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

2083 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 2083 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:

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

Polycyclic presentation:

2^9|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	#72272703
maximal subgroup of	#72272703
maximal normal subgroup of	#72272703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254230);