#1323208

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1687 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fghi,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=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, 1230404);