#500670

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

1222 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)(7,11)(8,12)(9,14)(10,16)(13,21)(15,22)(17,30)(18,27)(19,25)(20,26)(23,29)(24,31)(28,34)(32,36)(33,35);(1,2,4,3)(5,7)(6,11)(8,18)(9,17)(10,20)(12,27)(13,24)(14,30)(15,25)(16,26)(19,22)(21,31)(23,32)(28,35)(29,36)(33,34);(6,12)(7,17)(8,21)(9,22)(11,24)(13,15)(14,28)(16,29)(18,32)(19,20)(23,34)(25,36)(26,35)(27,30);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=e,d^2=e,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, 407866);