#129233

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

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, 36429);