#6536145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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)(3,8)(4,6)(5,7)(17,18,21,25)(19,27,29,23)(20,31,32,40)(22,35,34,26)(24,36,39,33)(28,43,44,48)(30,38,46,42)(37,45,41,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,23,25,27)(20,28,32,44)(22,30,34,46)(24,37,39,41)(26,38,35,42)(31,48,40,43)(33,45,36,47);(10,13)(14,16)(17,20)(18,24)(19,28)(21,32)(22,36)(23,37)(25,39)(26,40)(27,41)(29,44)(30,47)(31,35)(33,34)(38,43)(42,48)(45,46);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47);(11,15)(14,16)(20,32)(24,39)(28,44)(31,40)(33,36)(37,41)(43,48)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443341);