#492295

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=dei,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399491);