#6718591

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6625787);