#6871618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(10,13)(11,15)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,33,37,42,34,26,35,41)(30,39,46,47,44,43,45,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31)(18,24,27,29)(20,30,34,44)(22,32,23,36)(25,38,28,40)(26,39,33,43)(35,45,37,46)(41,47,42,48);(10,13)(14,16)(18,25)(22,23)(24,38)(26,41)(27,28)(29,40)(32,36)(33,42)(35,37)(39,47)(43,48)(45,46);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,37)(24,39)(25,41)(27,33)(28,42)(29,43)(31,44)(32,45)(36,46)(38,47)(40,48);(1,3,4,7)(2,6,5,8)(19,31)(20,34)(24,29)(26,33)(32,36)(35,37)(38,40)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

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, 6778814);