#500717

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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 ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1043 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)(7,19)(8,18)(9,16)(10,14)(11,28)(12,15)(13,22)(17,26)(20,33)(21,31)(23,36)(24,32)(25,34)(27,35)(29,30);(1,2,4,3)(5,7)(6,13)(8,21)(9,22)(10,27)(11,20)(12,29)(14,32)(15,31)(16,19)(17,33)(18,30)(23,26)(24,25)(28,36)(34,35);(6,14)(7,20)(8,12)(10,26)(11,25)(13,30)(15,28)(16,34)(17,18)(19,31)(21,27)(22,23)(24,29)(33,36);(1,3,4,2)(5,8,9,12)(6,15,16,18)(7,21,22,29)(10,11,25,26)(13,31,19,30)(14,17,34,28)(20,24,23,27)(32,33,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407913);