#611548

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

2136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

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, 518744);