#6848649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1633 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)(19,24)(20,26)(21,28)(22,27)(23,29)(25,34)(30,39)(31,46)(32,41)(33,44)(35,45)(36,40)(37,43)(38,48)(42,47);(1,3,4,8)(2,5,6,7)(11,12)(14,15)(17,19)(18,24)(20,33)(21,36)(22,31)(23,32)(25,42)(26,44)(27,46)(28,40)(29,41)(30,35)(34,47)(37,48)(38,43)(39,45);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,27)(28,45)(29,43)(31,47)(32,33)(36,48)(39,46)(40,44)(41,42);(2,6)(5,7)(17,20)(18,26)(19,30)(21,37)(22,34)(23,35)(24,39)(25,27)(28,43)(29,45)(31,48)(32,33)(36,47)(38,46)(40,42)(41,44);(9,12,13,11)(10,15,16,14)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,40,41,46)(25,26,43,45)(30,47,33,48)(38,44,42,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755845);