#147078

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

742 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)(5,6,11,16)(7,9)(8,19,23,35)(10,26,20,15)(12,14)(13,28,30,22)(17,32)(18,27)(21,33)(24,34)(25,31)(29,36);(2,4)(5,7,12,16,11,14,9,6)(8,17,24,30,23,29,21,13)(10,18,27,26,20,31,25,15)(19,22,33,36,35,28,34,32);(5,8,20,28,11,23,10,22)(6,13,26,35,16,30,15,19)(7,17,31,34,14,29,18,33)(9,21,27,36,12,24,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54274);