#7041602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=fi,e^2=i,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6948798);