#1267741

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15763	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=fh,d^2=f,e^2=h,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, 1174937);