#490344

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 673 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397540);