#492270

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
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

448 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 114 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)(19,27)(20,23)(21,33)(22,36)(24,29)(25,35)(26,32)(28,43)(30,45)(31,46)(34,42)(37,41)(38,44)(39,48)(40,47);(1,3)(2,6)(4,11)(5,10)(7,16)(8,15)(9,13)(12,14)(18,23)(24,38)(25,39)(26,36)(27,43)(33,46)(40,42)(41,45);(3,10)(6,15)(9,16)(11,12)(17,19)(18,24)(20,28)(21,34)(22,37)(23,38)(25,41)(26,42)(27,33)(29,35)(30,32)(31,47)(36,40)(39,45)(43,46)(44,48);(3,11)(4,14)(6,16)(7,13)(9,15)(10,12)(19,28)(24,38)(27,43)(29,44)(30,37)(34,47)(40,42)(41,45)

Polycyclic presentation:

2^9|a^2=g,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=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399466);