#6474682

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,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, 6381878);