#136265

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	C₂×#469	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 422 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313410, #72313765, #72313774
maximal normal subgroup of	#72313410, #72313765, #72313774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43461);