#6858629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2657	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6765825);