#6434536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(17,18,23,28)(19,29,31,45)(20,33,30,46)(21,26,35,43)(22,37,40,27)(24,38,42,32)(25,39,41,34)(36,47,48,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,42)(20,21,30,35)(22,38,40,32)(25,26,41,43)(27,45,37,29)(33,47,46,44)(34,48,39,36);(11,15)(13,16)(17,20,23,30)(18,25,28,41)(19,21,31,35)(22,39,40,34)(24,26,42,43)(27,46,37,33)(29,47,45,44)(32,48,38,36);(10,14)(13,16)(17,21)(18,26)(19,30)(20,31)(22,36)(23,35)(24,41)(25,42)(27,44)(28,43)(29,46)(32,39)(33,45)(34,38)(37,47)(40,48);(1,3,4,7)(2,6,5,8)(19,31)(21,35)(24,42)(26,43)(29,45)(32,38)(36,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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,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, 6341732);