#491457

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	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,32)(19,33,35,27)(20,40,36,26)(21,29,41,43)(22,28,34,47)(23,31,24,30)(37,46,42,45)(38,48,39,44);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,34,42,23,38,21,36,25,35,22,37,24,39,41,20)(18,26,43,48,30,45,28,33,32,40,29,44,31,46,47,27);(2,7)(6,14)(8,12)(9,15)(17,20,41,39,24,37,22,35,25,36,21,38,23,42,34,19)(18,27,47,46,31,44,29,40,32,33,28,45,30,48,43,26);(19,35)(20,36)(26,40)(27,33)(37,42)(38,39)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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, 398653);