#7091111

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,28,23,27,36,38)(19,29,31,26,32,42,21,24)(20,25,35,40,34,41,46,47)(30,43,45,39,44,48,33,37);(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,27,26)(20,30,34,44)(21,36,31,22)(25,37,41,39)(28,42,38,29)(33,46,45,35)(40,48,47,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,37)(26,39)(27,41)(28,40)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(11,16)(13,15)(17,21,22,19,23,31,36,32)(18,26,28,42,27,24,38,29)(20,33,35,30,34,45,46,44)(25,39,40,48,41,37,47,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,34)(24,26)(28,38)(29,42)(30,44)(33,45)(35,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,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, 6998307);