#489230

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	#61406	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

729 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396426);