#500674

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407870);