#686259

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593455);