#1907615

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1810 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,12,15,14)(18,20)(22,24)(27,31)(30,32);(1,2,4,6)(3,5,7,8)(9,11,13,15)(10,14,16,12)(19,23)(22,24)(26,29)(30,32);(3,7)(5,8)(9,12,13,14)(10,15,16,11)(17,18,21,20)(19,22,23,24)(25,26,28,29)(27,30,31,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,23)(18,22,20,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,21,18)(19,24,23,22)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1814811);