#950630

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2142 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)(9,10,13,16)(11,15,12,14)(17,18,20,22)(19,23,24,21)(26,28)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,22)(21,24)(25,26,29,28)(27,31,32,30);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,12,15)(17,19,20,24)(18,21,22,23)(25,26,29,28)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)(25,27,29,32)(26,30,28,31);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 857826);