#499145

order	512
nilpotency class	4
isoclinism	isoclinic class of #498886
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1147 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)(5,6,11,7)(8,18,21,28)(9,16,23,13)(10,17,25,14)(12,24,15,20)(19,32,34,31)(22,27,36,30)(26,33,29,35);(1,2)(3,4)(5,7,11,6)(8,15,21,12)(9,13,23,16)(10,14,25,17)(18,20,28,24)(19,26,34,29)(22,30,36,27)(31,35,32,33);(1,3,4,2)(5,8)(6,12)(7,15)(9,19)(10,24)(11,21)(13,26)(14,28)(16,29)(17,18)(20,25)(22,35)(23,34)(27,32)(30,31)(33,36);(5,9,11,23)(6,13,7,16)(8,19,21,34)(10,22,25,36)(12,26,15,29)(14,27,17,30)(18,31,28,32)(20,33,24,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,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, 406341);