#2142143

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,e^2=f,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, 2049339);