#7101868

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=fg,e^2=gi,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, 7009064);