#6484574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,28,23,29,21,27)(19,30,32,40,33,43,37,24)(20,26,34,41,35,42,36,25)(31,38,45,47,46,48,44,39);(2,5)(6,8)(10,14)(11,12)(17,19,23,33)(18,24,29,40)(20,31,35,46)(21,32,22,37)(25,38,41,48)(26,39,42,47)(27,30,28,43)(34,44,36,45);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,31)(21,34)(22,36)(23,35)(24,38)(26,27)(28,42)(29,41)(30,39)(32,44)(33,46)(37,45)(40,48)(43,47);(9,12,13,11)(10,16,14,15)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,39)(25,27)(28,41)(29,42)(30,38)(32,45)(33,46)(37,44)(40,47)(43,48);(10,14)(15,16)(17,21,23,22)(18,27,29,28)(19,32,33,37)(20,34,35,36)(24,30,40,43)(25,26,41,42)(31,44,46,45)(38,39,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391770);