#1294102

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,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, 1201298);