#6653304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,29,23)(20,31,33,42)(22,36,35,26)(24,37,41,34)(28,44,32,48)(30,39,46,43)(38,45,40,47);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,39)(28,33)(31,44)(34,47)(35,46)(36,43)(37,45)(38,41)(42,48);(11,16)(13,15)(17,20,21,33)(18,24,25,41)(19,28,29,32)(22,37,35,34)(23,38,27,40)(26,42,36,31)(30,47,46,45)(39,44,43,48);(2,3)(6,7)(10,14)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,38)(25,27)(26,39)(31,48)(32,33)(34,45)(35,46)(36,43)(37,47)(40,41)(42,44);(1,4,5,8)(2,6,3,7)(19,29)(23,27)(28,32)(30,46)(38,40)(39,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,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, 6560500);