#489533

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 629 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396729);