#6444328

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
prime radical	#60644	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2653 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)(5,6,12,18,11,7,10,15)(8,16,24,27,20,13,21,30)(9,14,25,31,23,17,22,28)(19,29,34,35,32,26,33,36);(5,7,11,6)(8,16,20,13)(9,17,23,14)(10,15,12,18)(19,26,32,29)(21,27,24,30)(22,28,25,31)(33,36,34,35);(1,2,3,4)(5,8,12,24,11,20,10,21)(6,13,18,30,7,16,15,27)(9,19,25,34,23,32,22,33)(14,26,31,36,17,29,28,35);(5,9,10,22,11,23,12,25)(6,14,15,28,7,17,18,31)(8,19,21,33,20,32,24,34)(13,26,27,35,16,29,30,36);(8,20)(13,16)(19,32)(21,24)(26,29)(27,30)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 6351524);