#6536870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(17,18,22,26)(19,27,30,41)(20,32,33,24)(21,35,28,25)(23,36,38,31)(29,43,44,40)(34,39,42,47)(37,45,46,48);(9,10)(11,14)(12,13)(15,16)(17,19,28,36,22,30,21,31)(18,23,35,41,26,38,25,27)(20,29,42,48,33,44,34,45)(24,37,47,43,32,46,39,40);(2,6)(7,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,39)(26,32)(27,40)(28,42)(30,44)(31,45)(35,47)(36,48)(38,46)(41,43);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,33)(24,32)(29,44)(34,42)(37,46)(39,47)(40,43)(45,48);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(19,30)(23,38)(27,41)(29,44)(31,36)(37,46)(40,43)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=eh

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, 6444066);