#129743

order	512
nilpotency class	4
isoclinism	isoclinic class of #129741
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#534	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

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

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, 36939);