#6524261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,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, 6431457);