#2805741

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2712937);