#492268

order	512
nilpotency class	4
isoclinism	isoclinic class of #492224
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×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 435 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399464);