#136018

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
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	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 440 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)(5,6,9,16,11,17,10,15)(7,18,21,29,22,32,20,13)(8,23,25,31,26,12,24,14)(19,30,34,36,35,28,33,27);(1,3,4,2)(6,12,17,23)(8,24)(9,10)(13,27,29,36)(14,16,31,15)(18,28,32,30)(19,33)(20,21)(25,26)(34,35);(5,7)(6,13)(8,19)(9,21)(10,20)(11,22)(12,27)(14,30)(15,32)(16,18)(17,29)(23,36)(24,33)(25,34)(26,35)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313510, #72313521, #72313566
maximal normal subgroup of	#72313510, #72313521, #72313566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43214);