#347041

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

1905 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1905 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);(1,2,8,14)(3,11,20,42)(4,12,24,45)(5,26,23,10)(6,31,28,58)(7,34,9,15)(13,33,44,22)(16,47,50,61)(17,39,53,64)(18,54,51,40)(19,43,35,60)(21,55,25,46)(27,57,56,36)(29,37,30,48)(32,41,59,63)(38,49,62,52);(1,3,9,19)(2,11,15,43)(4,17,25,52)(5,27,30,18)(6,32,22,16)(7,35,8,20)(10,36,37,40)(12,39,46,38)(13,47,31,41)(14,42,34,60)(21,49,24,53)(23,56,29,51)(26,57,48,54)(28,59,33,50)(44,61,58,63)(45,64,55,62);(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|b^2=h,c^2=i,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	#72272595
maximal subgroup of	#72272561, #72272595
maximal normal subgroup of	#72272561, #72272595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254237);