#6602947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fi,d^2=fi,e^2=i,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,g^c=gi,e^d=ehi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6510143);