#499556

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=i,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, 406752);