#7026435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,28,40)(20,24,32,39)(21,34,29,26)(23,35,36,31)(30,41,42,48)(33,45,43,38)(37,46,47,44);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(19,28)(23,36)(27,40)(30,42)(31,35)(37,47)(41,48)(44,46);(9,11)(10,13)(12,16)(14,15)(17,19,29,35,22,28,21,31)(18,23,34,40,25,36,26,27)(20,30,43,46,32,42,33,44)(24,37,45,48,39,47,38,41);(2,5)(6,8)(11,16)(13,15)(17,20)(18,24)(19,30)(21,33)(22,32)(23,37)(25,39)(26,38)(27,41)(28,42)(29,43)(31,44)(34,45)(35,46)(36,47)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(20,32)(23,36)(26,34)(27,40)(30,42)(33,43)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6933631);