#215137

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1036 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 210 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)(3,7)(4,6)(5,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,29,33,30,32,28,37)(27,31,36,38,34,39,35,40);(1,3,4,8)(2,5,6,7)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(3,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(26,32)(27,34)(28,29)(38,40);(2,6)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,29)(26,33,32,37)(27,35,34,36)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di,g^c=gh

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, 122333);