#492816

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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,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, 400012);