#7094849

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 7002045);