#347035

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

1958 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|b^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

maximal subgroup of	#72272561
maximal normal subgroup of	#72272561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254231);