#135233

order	512
nilpotency class	4
isoclinism	isoclinic class of #129414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=chi,e^c=ei,f^c=fgi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42429);