#6761600

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

1200 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=hi,d^2=f,e^2=f,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, 6668796);