#6475244

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

1363 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,26)(19,25,29,23)(20,31,33,47)(22,36,35,27)(24,39,41,34)(28,43,32,46)(30,38,45,42)(37,44,40,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,26)(27,38)(28,33)(31,43)(34,48)(35,45)(36,42)(37,41)(39,44)(46,47);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,34,21,33,35,39)(18,24,27,31,26,41,36,47)(19,28,30,44,29,32,45,48)(23,37,38,46,25,40,42,43);(11,15)(13,16)(17,19)(18,25)(20,32)(21,29)(22,30)(23,26)(24,37)(27,42)(28,33)(31,46)(34,48)(35,45)(36,38)(39,44)(40,41)(43,47);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,26)(19,29)(24,41)(27,36)(28,32)(30,45)(31,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,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, 6382440);