#2039983

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3162 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,23,24,21)(25,26,29,27)(28,31,32,30);(1,3,5,8)(2,6,7,4)(11,15)(13,16)(17,18,20,22)(19,21,24,23)(26,27)(28,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,24)(22,23)(25,27,29,26)(28,31,32,30);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,24)(22,23)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1947179);