#2806529

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 2097152
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

3710 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)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24);(1,2,3,4)(10,12)(15,20)(16,19)(23,28)(24,27);(9,10)(11,12)(13,15,18,20)(14,19,22,16)(17,23,26,28)(21,27,25,24);(1,2,3,4)(5,6)(7,8)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23);(1,2)(3,4)(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=fg,e^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, 2713725);