#405908

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313104);