#491458

order	512
nilpotency class	4
isoclinism	isoclinic class of #491437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398654);