#1079011

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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

2260 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)(9,10,13,16)(11,15,12,14)(18,21)(19,20)(25,26,29,31)(27,28,32,30);(9,11,13,12)(10,14,16,15)(17,18,22,21)(19,23,20,24)(25,27,29,32)(26,30,31,28);(9,11,13,12)(10,15,16,14)(17,19,22,20)(18,23,21,24)(25,28,29,30)(26,27,31,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,15,16,14)(17,20,22,19)(18,23,21,24)(25,26)(27,28)(29,31)(30,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,21,22,18)(19,23,20,24)(25,28)(26,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=f,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 986207);