#492155

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

389 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)(7,15)(8,13)(9,16)(10,14)(17,18,21,24)(19,28,30,45)(20,26,23,38)(22,35,34,46)(25,41,39,31)(27,33,40,37)(29,44,32,48)(36,43,47,42);(1,2,5,8)(3,9,11,15)(4,13,12,6)(7,14,16,10)(18,24)(25,39)(26,38)(27,40)(28,45)(35,46)(42,43)(44,48);(17,19,21,30)(18,25,24,39)(20,32,23,29)(22,36,34,47)(26,43,38,42)(27,44,40,48)(28,46,45,35)(31,33,41,37);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(17,20,21,23)(18,26,24,38)(19,29,30,32)(22,33,34,37)(25,42,39,43)(27,35,40,46)(28,44,45,48)(31,47,41,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gi,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, 399351);