#6696860

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,27)(19,24,32,41)(20,33,36,25)(21,26,23,28)(29,45,47,38)(30,39,31,40)(34,48,46,42)(35,44,37,43);(9,10)(11,15)(12,16)(13,14)(17,19,22,32)(18,24,27,41)(20,34,36,46)(21,30,23,31)(25,42,33,48)(26,39,28,40)(29,37,47,35)(38,44,45,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,38)(26,43)(27,33)(28,44)(30,46)(31,34)(32,47)(39,48)(40,42)(41,45);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,28)(19,30,32,31)(20,35,36,37)(24,39,41,40)(25,43,33,44)(29,46,47,34)(38,48,45,42);(10,14)(15,16)(19,31)(21,23)(24,40)(26,28)(29,34)(30,32)(35,37)(38,42)(39,41)(43,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^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, 6604056);