#148529

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
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

662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250002, #72250005, #72250032, …
maximal normal subgroup of	#72250002, #72250005, #72250032, …
maximal quotient of	#72308716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55725);