#759728

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(6,8)(13,14,18,16)(15,20,17,19);(9,10,11,12)(13,15,18,17)(14,19,16,20);(5,6)(7,8)(9,10)(11,12)(15,17)(19,20);(1,2)(3,4)(5,6)(7,8)(13,16,18,14)(15,20,17,19);(13,17,18,15)(14,20,16,19)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 666924);