#6443055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350251);