#473950

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

880 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381146);