#6443053

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

2562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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, 6350249);