#492755

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
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

397 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399951);