#1126159

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1716 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,22,23,20)(25,26,29,28)(27,30,31,32);(1,2,4,6)(3,5,7,8)(17,19,21,23)(18,22,24,20)(27,31)(30,32);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27,29,31)(26,30,28,32);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,21,22)(18,23,24,19);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 1033355);