#7092643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1279 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)(17,18,21,27,22,26,34,23)(19,24,30,38,29,39,45,35)(20,31,32,36,33,25,40,41)(28,42,43,46,44,37,47,48);(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(21,34)(24,35)(26,27)(28,43)(30,45)(33,40)(36,41)(38,39)(44,47)(46,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,26,39)(20,28,33,44)(21,30,34,45)(23,35,27,38)(25,37,31,42)(32,43,40,47)(36,46,41,48);(11,16)(13,15)(17,20,21,32,22,33,34,40)(18,25,27,41,26,31,23,36)(19,28,30,43,29,44,45,47)(24,37,38,48,39,42,35,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,29)(23,27)(25,31)(28,44)(30,45)(36,41)(43,47)

Polycyclic presentation:

2^9|a^2=fg,c^2=ghi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999839);