#6524405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(17,18,22,26)(19,27,30,41)(20,32,33,24)(21,35,28,25)(23,36,38,31)(29,43,44,40)(34,39,42,47)(37,45,46,48);(9,10)(11,14)(12,13)(15,16)(17,19,28,36,22,30,21,31)(18,23,35,41,26,38,25,27)(20,29,42,48,33,44,34,45)(24,37,47,43,32,46,39,40);(2,5)(6,8)(10,13)(11,15)(17,20,22,33)(18,24,26,32)(19,29,30,44)(21,34,28,42)(23,37,38,46)(25,39,35,47)(27,40,41,43)(31,45,36,48);(9,11,12,15)(10,14,13,16)(19,30)(23,38)(27,41)(29,44)(31,36)(37,46)(40,43)(45,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(20,33)(23,38)(24,32)(27,41)(31,36)(34,42)(39,47)

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,d^b=di,e^b=ehi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431601);