#7074713

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6981909);