#489247

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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

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

normal subgroups only

This group has 728 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)(19,30)(20,28)(21,36)(22,26)(23,29)(24,32)(25,31)(27,38)(33,44)(34,43)(35,40)(37,46)(39,45)(41,48)(42,47);(4,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,31)(21,37)(22,34)(23,33)(25,40)(26,41)(27,43)(28,46)(29,42)(30,36)(32,38)(35,47)(39,48)(44,45);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,19)(18,25)(20,34)(21,32)(22,31)(23,33)(24,40)(26,44)(27,42)(28,30)(29,43)(35,48)(36,46)(37,38)(39,47)(41,45);(17,20,23,22)(18,26,29,28)(19,31,33,34)(21,35,39,38)(24,41,42,46)(25,30,43,44)(27,45,40,36)(32,37,47,48)

Polycyclic presentation:

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

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, 396443);