#7030339

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,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=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, 6937535);