#414814

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322010);