#147325

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

702 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)(7,17)(8,14)(9,25)(10,15)(11,16)(12,19)(13,23)(18,29)(20,33)(21,27)(22,32)(24,30)(26,31)(28,34)(35,36);(5,7,10,21,11,22,9,20)(6,12,15,28,16,29,25,36)(8,18,23,34,24,19,26,35)(13,27,30,17,31,33,14,32);(1,3,4,2)(5,8)(6,13)(7,18)(9,26)(10,23)(11,24)(12,27)(14,25)(15,30)(16,31)(17,28)(19,22)(20,35)(21,34)(29,33)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^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, 54521);