#499136

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1167 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:

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

Polycyclic presentation:

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

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, 406332);