#6799356

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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)(25,26,29,34)(27,36,38,31)(28,40,42,33)(30,32,44,35)(37,46,41,49)(39,47,52,45)(43,50,54,48)(51,55,53,56);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,41)(29,38)(30,39)(32,45)(33,49)(34,36)(35,47)(37,42)(40,46)(43,53)(44,52)(48,56)(50,55)(51,54);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,45)(33,48)(34,35)(36,47)(39,52)(40,50)(43,54)(46,55)(49,56)(51,53);(10,16)(14,17)(15,22)(21,23)(25,28)(26,33)(27,37)(29,42)(30,43)(31,46)(32,48)(34,40)(35,50)(36,49)(38,41)(39,51)(44,54)(45,55)(47,56)(52,53);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,38)(28,42)(31,36)(33,40)(39,52)(43,54)(45,47)(48,50)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6706552);