#6728987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42760	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,24,31,41)(20,33,36,25)(21,26,38,23)(28,45,46,40)(29,32,42,30)(34,48,47,43)(35,39,44,37);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,34)(21,29)(22,31)(23,30)(25,43)(26,32)(27,41)(28,37)(33,48)(35,45)(36,47)(38,42)(39,46)(40,44);(10,14)(15,16)(17,20,22,36)(18,25,27,33)(19,28,31,46)(21,35,38,44)(23,39,26,37)(24,40,41,45)(29,43,42,48)(30,34,32,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,35)(22,38)(23,27)(24,32)(25,37)(28,43)(30,41)(31,42)(33,39)(34,45)(36,44)(40,47)(46,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,30)(21,38)(23,26)(24,29)(28,34)(31,32)(35,44)(37,39)(40,43)(41,42)(45,48)(46,47)

Polycyclic presentation:

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

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, 6636183);