#6431735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338931);