#6444325

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2707 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi

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, 6351521);