#385703

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1104 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)(10,13)(11,15)(17,18,22,23,21,25,19,26)(20,28,30,32,29,24,27,31);(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,28,32);(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,25,28)(19,27,22,30)(23,31,26,32);(1,3,4,2)(10,13)(14,16)(20,29)(24,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=ef,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, 292899);