#129724

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 948 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi,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, 36920);