#6848605

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6755801);