#6622954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(4,6)(5,8)(10,14)(15,16)(17,18,22,26)(19,23,28,36)(20,30,32,45)(21,34,31,25)(24,38,39,33)(27,40,42,48)(29,44,41,37)(35,46,47,43);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,37)(26,36)(30,40)(31,41)(32,42)(33,43)(34,44)(38,46)(39,47)(45,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,31,38,22,32,21,33)(18,24,34,45,26,39,25,30)(19,27,41,46,28,42,29,43)(23,35,44,48,36,47,37,40);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(19,28)(23,36)(27,42)(29,41)(35,47)(37,44)(40,48)(43,46);(9,12,13,11)(10,16,14,15)(20,32)(24,39)(27,42)(30,45)(33,38)(35,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,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, 6530150);