#6848612

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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fhi,d^2=i,e^2=fhi,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, 6755808);