#7090362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1643 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)(17,18,23,27)(19,29,21,26)(20,33,35,25)(22,38,30,28)(24,37,40,32)(31,44,34,42)(36,43,45,47)(39,46,41,48);(9,10)(11,15)(12,14)(13,16)(17,19,30,37,23,21,22,32)(18,24,38,26,27,40,28,29)(20,34,45,46,35,31,36,48)(25,41,47,42,33,39,43,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,34)(22,36)(23,35)(24,39)(26,44)(27,33)(28,43)(29,42)(30,45)(32,46)(37,48)(38,47)(40,41);(11,16)(13,15)(17,21,23,19)(18,26,27,29)(20,31,35,34)(22,32,30,37)(24,28,40,38)(25,42,33,44)(36,48,45,46)(39,47,41,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,35)(24,40)(26,29)(28,38)(31,34)(36,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997558);