#7199962

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1227 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=gi,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 7107158);