#7092398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999594);