#1379073

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×#2682	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3272 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=fgi,d^2=f,e^2=gi,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, 1286269);