#492158

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=g,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,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399354);