#148145

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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

758 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)(5,6,11,17)(7,19,23,34)(8,18,12,15)(9,14,10,26)(13,28,29,21)(16,24,32,25)(20,33,27,35)(22,31,36,30);(5,7,8,21,11,23,12,28)(6,13,15,19,17,29,18,34)(9,20,24,36,10,27,25,22)(14,30,16,33,26,31,32,35);(6,14)(7,20)(8,12)(9,25)(10,24)(13,29)(15,32)(16,18)(17,26)(21,22)(23,27)(28,36)(30,35)(31,33)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250362, #72250367, #72250390, …
maximal normal subgroup of	#72250362, #72250367, #72250390, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55341);