#6718357

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,31)(21,25)(22,26)(24,33)(27,45)(28,43)(29,39)(30,38)(32,48)(34,44)(35,42)(36,41)(37,46)(40,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,21,29)(18,23,25,39)(20,32,33,47)(22,28,35,30)(24,40,31,48)(26,43,42,38)(27,36,46,34)(37,44,45,41);(10,13)(14,16)(17,20,21,33)(18,24,25,31)(19,27,29,46)(22,36,35,34)(23,37,39,45)(26,44,42,41)(28,32,30,47)(38,48,43,40);(2,6)(5,7)(10,13)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,37)(28,43)(29,39)(30,38)(31,33)(32,40)(34,41)(35,42)(36,44)(45,46)(47,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(22,35)(23,38)(24,31)(27,32)(29,30)(34,36)(37,48)(39,43)(40,45)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6625553);