#6693490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(11,13)(12,15)(14,16)(17,18)(19,28)(20,33)(21,27)(22,25)(23,26)(24,32)(29,38)(30,39)(31,45)(34,44)(35,48)(36,41)(37,42)(40,47)(43,46);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,30)(21,35)(22,32)(23,29)(25,28)(26,39)(27,43)(31,37)(33,38)(34,47)(36,46)(40,42)(41,48)(44,45);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(18,25)(19,29)(20,23)(24,38)(27,41)(28,39)(30,32)(31,46)(34,37)(35,47)(40,43)(45,48);(9,12,13,16)(10,14,11,15)(17,20,22,23)(18,26,25,33)(19,30,32,29)(21,34,36,37)(24,39,28,38)(27,42,41,44)(31,35,47,46)(40,48,45,43);(17,21,22,36)(18,27,25,41)(19,31,32,47)(20,34,23,37)(24,40,28,45)(26,42,33,44)(29,46,30,35)(38,43,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600686);