#7090364

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1670 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)(17,18,22,25)(19,27,32,39)(20,33,35,24)(21,37,28,26)(23,30,40,31)(29,45,34,43)(36,42,44,48)(38,46,41,47);(9,10)(11,15)(12,14)(13,16)(17,19,28,30,22,32,21,31)(18,23,37,39,25,40,26,27)(20,34,44,46,35,29,36,47)(24,41,48,43,33,38,42,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,35)(18,24,25,33)(19,29,32,34)(21,36,28,44)(23,38,40,41)(26,42,37,48)(27,43,39,45)(30,47,31,46);(11,16)(13,15)(19,30)(20,35)(21,28)(23,39)(24,33)(26,37)(27,40)(29,46)(31,32)(34,47)(38,43)(41,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,25)(20,35)(23,40)(26,37)(27,39)(29,34)(36,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,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, 6997560);