#1260066

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5085 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)(13,14,18,15)(16,21,24,19)(17,20,26,22)(23,28,25,27);(10,12)(14,15)(19,21)(20,22)(27,28);(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(5,6)(7,8)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(1,2)(3,4)(5,6)(7,8)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=f,e^2=f,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, 1167262);