#6860223

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6767419);