#3067298

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(18,20)(19,24)(26,29)(30,31);(1,3,6,4)(2,5,8,7)(11,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26,28,29)(27,31,32,30);(1,3,6,4)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,7,8,5)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,23,24,22)(25,27,28,32)(26,30,29,31);(1,5,6,7)(2,4,8,3)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=gi,c^2=ghi,d^2=f,e^2=ghi,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, 2974494);