#6731043

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,27,31,23)(20,32,33,25)(22,26,35,38)(28,36,30,40)(29,37,43,41)(34,45,46,39)(42,48,44,47);(10,13)(11,15)(17,19,21,31)(18,23,24,27)(20,29,33,43)(22,28,35,30)(25,37,32,41)(26,40,38,36)(34,42,46,44)(39,48,45,47);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(22,35)(23,36)(25,32)(27,40)(29,42)(30,31)(34,46)(37,47)(41,48)(43,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,32)(26,39)(27,41)(28,42)(30,44)(31,43)(35,46)(36,47)(38,45)(40,48);(1,4,5,3)(2,8,6,7)(19,30)(20,33)(22,35)(23,36)(25,32)(26,38)(27,40)(28,31)(29,42)(37,48)(41,47)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,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, 6638239);