#492013

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
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×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

435 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:

(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(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);(1,2,5,9)(3,7,11,15)(4,8,12,6)(10,14,16,13)(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);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,20)(18,25)(19,29)(21,32)(22,36)(23,38)(24,39)(26,42)(27,43)(28,45)(30,41)(31,35)(33,37)(34,48)(40,47)(44,46);(2,8)(4,12)(6,9)(7,14)(10,16)(13,15)(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=i,b^2=gi,c^2=g,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,d^b=dei,e^b=eg,f^c=fh,h^c=hi,e^d=eg

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, 399209);