#6437976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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

4570 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,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, 6345172);