#6536091

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(17,18,22,26)(19,23,28,37)(20,30,31,40)(21,33,35,25)(24,34,39,32)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,26,37)(20,27,31,42)(21,29,35,44)(24,36,39,47)(25,38,33,45)(30,41,40,48)(32,43,34,46);(10,13)(14,16)(17,20,22,31)(18,24,26,39)(19,27,28,42)(21,34,35,32)(23,36,37,47)(25,40,33,30)(29,46,44,43)(38,48,45,41);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(20,31)(24,39)(27,42)(30,40)(32,34)(36,47)(41,48)(43,46);(11,15)(14,16)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,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, 6443287);