#136011

order	512
nilpotency class	4
isoclinism	isoclinic class of #136011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43207);