#499704

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406900);