#148550

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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)(7,18,22,33)(8,24)(9,27)(10,14)(11,26)(12,17)(13,20,34,29)(15,28)(16,30)(19,32,35,21)(23,36,31,25);(5,7)(6,13)(8,25)(9,21)(10,29)(11,31)(12,22)(14,36)(15,18)(16,19)(17,34)(20,24)(23,30)(26,35)(27,33)(28,32);(2,4)(5,8,9,26,11,15,10,6,12,14,28,16,30,27,24,17)(7,19,20,36,23,13,32,18,22,35,29,25,31,34,21,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250013, #72250032, #72250045, …
maximal normal subgroup of	#72250013, #72250032, #72250045, …
maximal quotient of	#72308716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55746);