#148479

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

752 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)(5,6,12,16,11,17,10,31)(7,18,14,8,22,24,21,25)(9,26,30,35,29,15,28,19)(13,32,23,27,33,34,36,20);(5,7,11,22)(6,13,17,33)(8,19,25,35)(9,27,29,20)(10,14,12,21)(15,24,26,18)(16,23,31,36)(28,34,30,32);(2,4)(5,8,9,23,11,25,29,36)(6,14,15,34,17,21,26,32)(7,19,20,16,22,35,27,31)(10,24,30,13,12,18,28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55675);