#2487467

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1866 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=f,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2394663);