#386146

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1542 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=dei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293342);