#327306

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
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

1008 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(2,5)(6,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19)(25,27)(26,30)(28,35)(29,34)(31,39)(32,37)(33,36)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19)(30,37)(38,39);(3,7)(6,8)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,38,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,d^c=de,f^c=fh

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, 234502);