#6437938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4572 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,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, 6345134);