#511621

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

450 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)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18)(19,28)(20,32)(21,26)(22,27)(23,25)(24,31)(29,44)(30,39)(33,47)(34,42)(35,41)(36,48)(37,40)(38,45)(43,46);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,15,14)(19,29)(20,23)(22,35)(24,38)(25,32)(27,41)(28,44)(30,46)(31,45)(33,34)(36,37)(39,43)(40,48)(42,47);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,31)(18,24,26,28)(20,30,23,37)(22,36,34,46)(25,39,32,40)(27,43,42,48)(29,33,45,35)(38,47,44,41);(3,7)(5,8)(11,16)(13,14)(17,20,21,23)(18,25,26,32)(19,30,31,37)(22,33,34,35)(24,39,28,40)(27,41,42,47)(29,36,45,46)(38,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418817);