#6818104

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725300);