#2814423

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
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3293 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=f,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, 2721619);