#500678

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 407874);