#136016

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 458 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) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^d=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, 43212);