#6825556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#24271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(11,14)(12,13)(15,16)(19,27)(20,32)(23,36)(24,39)(28,41)(31,37)(33,46)(34,40);(2,5)(3,7)(17,18,21,25)(19,28,27,41)(20,33,32,46)(22,35,29,26)(23,37,36,31)(24,40,39,34)(30,44,42,38)(43,47,48,45);(10,13)(14,16)(17,19,29,37,21,27,22,31)(18,23,35,41,25,36,26,28)(20,30,40,47,32,42,34,45)(24,38,46,48,39,44,33,43);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(20,32)(24,39)(30,42)(33,46)(34,40)(38,44)(43,48)(45,47);(1,3,4,7)(2,6,5,8)(17,20,29,40,21,32,22,34)(18,24,35,46,25,39,26,33)(19,30,37,47,27,42,31,45)(23,38,41,48,36,44,28,43)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6732752);