#492537

order	512
nilpotency class	4
isoclinism	isoclinic class of D₈×D₁₆
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	#19968	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

452 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399733);