#6868535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1246 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)(25,26,30,35,29,34,44,32)(27,36,39,46,38,31,45,48)(28,40,43,49,42,33,50,52)(37,47,53,56,41,51,54,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,45)(32,46,35,48)(33,51,40,47)(43,54,50,53)(49,56,52,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,46)(33,49)(34,35)(36,48)(39,45)(40,52)(43,50)(47,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,50)(31,47,36,51)(32,49,35,52)(39,53,45,54)(46,55,48,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(27,38)(28,42)(31,36)(33,40)(39,45)(43,50)(46,48)(49,52)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775731);