#7005064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^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=dfi,e^b=eh,d^c=dh,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, 6912260);