#491304

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398500);