#6434490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1676 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,31,39)(20,32,34,41)(21,26,22,35)(24,36,40,30)(25,37,42,33)(29,38,45,48)(43,44,47,46);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,38)(26,39)(27,40)(28,35)(32,43)(33,44)(34,45)(37,46)(41,47)(42,48);(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,42)(28,43)(30,44)(31,45)(32,35)(36,46)(39,47)(40,48);(10,14)(13,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,40,28)(25,41,42,32)(29,44,45,46)(38,47,48,43);(1,3,4,7)(2,6,5,8)(19,31)(24,40)(28,39)(29,45)(30,36)(38,48)(43,47)(44,46)

Polycyclic presentation:

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

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, 6341686);