#7144098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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)(11,15)(14,16)(17,18,21,26)(19,28,29,23)(20,25,32,40)(22,24,34,27)(30,41,42,36)(31,35,43,37)(33,38,45,39)(44,46,48,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,30)(21,29)(22,31)(24,37)(25,36)(26,28)(27,35)(32,42)(33,44)(34,43)(38,47)(39,46)(40,41)(45,48);(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(20,32)(22,34)(23,35)(25,39)(26,27)(28,37)(36,47)(38,40)(41,46)(44,48);(3,8)(5,7)(10,13)(14,16)(17,20,21,32)(18,25,26,40)(19,30,29,42)(22,33,34,45)(23,36,28,41)(24,38,27,39)(31,44,43,48)(35,46,37,47);(9,11,12,15)(10,14,13,16)(19,29)(20,32)(23,28)(25,40)(31,43)(33,45)(35,37)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7051294);