#499910

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

1388 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 407106);