#6468110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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

1610 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375306);