#411277

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,8)(4,7)(5,6)(10,13)(14,15)(17,18,22,37,23,27,34,28)(19,24,38,41,31,42,32,47)(20,33,21,25,35,26,36,43)(29,44,46,48,45,40,30,39);(2,6)(7,8)(9,10,12,13)(11,15,16,14)(17,19,22,32,23,31,34,38)(18,24,28,41,27,42,37,47)(20,29,21,30,35,45,36,46)(25,39,33,44,43,48,26,40);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,34,36,23,35,22,21)(18,25,37,26,27,43,28,33)(19,29,38,46,31,45,32,30)(24,39,47,40,42,48,41,44);(1,4,5,3)(2,7,6,8)(17,21,22,35,23,36,34,20)(18,26,28,25,27,33,37,43)(19,30,32,45,31,46,38,29)(24,40,41,39,42,44,47,48)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318473);