#499908

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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 ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1447 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)(7,18)(8,16)(9,24)(10,14)(11,17)(12,19)(13,20)(15,26)(21,32)(22,30)(23,28)(25,34)(27,33)(29,35)(31,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,28)(14,32)(15,29)(16,18)(17,30)(23,36)(24,34)(25,26)(27,35)(31,33);(2,4)(5,7)(6,13)(8,19)(9,25)(10,22)(11,21)(12,28)(14,31)(15,30)(16,34)(17,29)(18,24)(20,26)(23,35)(27,36)(32,33);(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,28)(14,32)(15,29)(16,18)(17,30)(23,36)(24,34)(25,26)(27,35)(31,33)

Polycyclic presentation:

2^9|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,e^b=ei,f^c=fh,h^c=hi,e^d=ei

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, 407104);