#237841

order	512
nilpotency class	3
isoclinism	isoclinic class of #232393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

992 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=h,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145037);