#6860092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767288);