#6821714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,29,31,24)(20,25,34,43)(21,37,38,27)(23,28,33,41)(30,45,47,39)(32,42,35,40)(36,44,46,48);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,26,29)(20,32,34,35)(21,23,38,33)(25,40,43,42)(27,28,37,41)(30,46,47,36)(39,48,45,44);(10,13)(14,16)(17,20,33,46,22,34,23,36)(18,25,41,48,26,43,28,44)(19,30,38,35,31,47,21,32)(24,39,37,42,29,45,27,40);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(20,34)(24,29)(27,37)(28,41)(30,47)(32,35)(36,46);(3,8)(5,7)(17,21,33,19,22,38,23,31)(18,27,41,24,26,37,28,29)(20,35,46,47,34,32,36,30)(25,42,48,45,43,40,44,39)

Polycyclic presentation:

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

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, 6728910);