#473947

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
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

894 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 186 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)(20,35,38,26,39,43,37,44)(32,46,47,45,48,41,36,42);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,24,34,23,21)(18,25,28,31,30,40,29,27)(20,36,37,48,39,47,38,32)(26,42,43,46,44,45,35,41);(3,7)(5,8)(10,13)(11,15)(17,20,24,39)(18,26,30,44)(19,32,34,48)(21,36,33,47)(22,38,23,37)(25,41,40,46)(27,42,31,45)(28,35,29,43);(9,11,12,15)(10,14,13,16)(17,21,23,34,24,33,22,19)(18,27,29,40,30,31,28,25)(20,32,38,47,39,48,37,36)(26,41,35,45,44,46,43,42)

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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381143);