#490339

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

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, 397535);