#501056

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408252);