#136191

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=gi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,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, 43387);