#2086549

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3381 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22)(23,28)(24,26)(25,27);(6,7)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,25,24)(22,26,27,28);(8,12)(10,11)(14,19)(16,18)(23,24)(26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,26)(25,27);(5,8,9,12)(6,10,7,11)(13,16,17,18)(14,15,19,20)(21,24,25,23)(22,26,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1993745);