#489232

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 697 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) = 256

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 396428);