#6651375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	#30413	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,32)(21,26)(22,24)(23,29)(25,34)(27,36)(30,38)(31,37)(33,41)(35,42)(39,43)(40,46)(44,48)(45,47);(1,2)(3,6)(4,5)(7,8)(11,12)(14,15)(17,19)(18,23)(20,33)(21,29)(22,31)(24,39)(25,41)(26,28)(27,37)(30,34)(32,38)(35,45)(36,43)(40,47)(42,48)(44,46);(9,11,13,12)(10,14,16,15)(18,24)(19,29)(22,36)(23,37)(25,40)(26,27)(28,39)(30,33)(32,42)(35,46)(38,47)(41,48);(2,5)(6,7)(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,30)(21,34)(22,35)(23,38)(24,40)(26,32)(27,42)(28,41)(29,33)(31,44)(36,46)(37,47)(39,48)(43,45);(1,3,4,8)(2,6,5,7)(10,16)(14,15)(20,34)(25,32)(30,33)(35,46)(38,41)(40,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,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, 6558571);