#6720272

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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)(11,16)(12,14)(13,15);(1,3)(2,5)(4,8)(6,7)(17,18,22,27)(19,28,31,41)(20,24,32,39)(21,25,35,26)(23,34,37,30)(29,42,43,47)(33,40,45,38)(36,46,48,44);(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,23,27,37)(20,29,32,43)(21,34,35,30)(24,36,39,48)(25,28,26,41)(33,46,45,44)(38,47,40,42);(2,6)(5,7)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,40)(26,38)(27,39)(28,42)(30,44)(31,43)(34,46)(35,45)(37,48)(41,47);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(18,25)(20,32)(21,35)(23,28)(24,38)(26,27)(29,43)(30,34)(36,47)(37,41)(39,40)(42,48)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=g,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, 6627468);