#7195094

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7102290);