#147323

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gh,d^b=dh,h^b=hi,e^c=ei,g^d=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, 54519);