#6442951

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2510 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,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, 6350147);