#7063960

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28,23,29,21,27)(19,24,32,41,33,42,31,40)(20,34,35,43,36,26,37,25)(30,44,45,48,46,39,47,38);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,29,42)(20,30,36,46)(21,31,22,32)(25,38,43,48)(26,39,34,44)(27,40,28,41)(35,45,37,47);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,28)(27,34)(29,43)(31,45)(32,47)(33,46)(39,41)(40,44)(42,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,37)(22,35)(23,36)(24,39)(25,28)(27,43)(29,34)(31,47)(32,45)(33,46)(38,41)(40,48)(42,44);(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,31,33,32)(20,35,36,37)(24,40,42,41)(25,34,43,26)(30,45,46,47)(38,44,48,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 6971156);