#6749566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,34)(21,28)(22,26)(23,27)(24,31)(25,36)(30,45)(32,40)(33,41)(35,48)(37,44)(38,43)(39,47)(42,46);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,29)(27,40)(28,41)(30,37)(34,45)(36,47)(38,46)(39,43)(44,48);(11,15)(13,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,43)(27,34)(28,44)(29,42)(31,35)(32,46)(33,47)(40,48)(41,45);(1,4,6,5)(2,7,3,8)(18,26)(19,31)(25,43)(27,28)(30,35)(32,33)(34,44)(46,47);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,36,37,38)(24,40,29,41)(25,34,43,44)(30,46,35,47)(39,48,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6656762);