#148385

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224865, #72224939, #72226822, …
maximal normal subgroup of	#72224865, #72224939, #72226822, …
maximal quotient of	#72224864, #72237233, #72280188

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55581);