#2076489

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2146 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)(3,7,8,5)(9,10,14,16)(11,13,12,15)(17,18,23,21)(19,24,22,20)(25,26)(27,31)(28,29)(30,32);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,15,16,13)(17,19,23,22)(18,24,21,20)(26,29)(27,32);(9,11,14,12)(10,13,16,15)(17,20)(18,22)(19,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,15,16,13)(17,21,23,18)(19,24,22,20);(1,3)(2,5)(4,8)(6,7)(9,13,14,15)(10,11,16,12)(17,22,23,19)(18,24,21,20)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fghi,d^2=fhi,e^2=fhi,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, 1983685);