#148610

order	512
nilpotency class	4
isoclinism	isoclinic class of #137661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ehi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320962, #72321073, #72321131, …
maximal normal subgroup of	#72320962, #72321073, #72321131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55806);