#6599427

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,30)(25,36)(29,45)(31,44)(32,40)(34,48)(35,43)(38,42)(39,47)(41,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,27,28)(20,34,35,29)(21,31,23,32)(25,41,42,39)(26,40,37,44)(33,45,43,48)(36,47,38,46);(11,12)(14,16)(17,20,22,35)(18,25,27,42)(19,29,30,34)(21,36,23,38)(24,39,28,41)(26,43,37,33)(31,46,32,47)(40,45,44,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,30)(20,35)(24,28)(25,42)(31,32)(33,43)(36,38)(40,44);(11,12)(14,16)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,36,35,38)(24,40,28,44)(25,43,42,33)(29,46,34,47)(39,45,41,48)

Polycyclic presentation:

2^9|b^2=h,c^2=i,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=ehi,d^c=dh,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, 6506623);