#6629037

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4424 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:

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

Polycyclic presentation:

2^9|b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,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, 6536233);