#2945919

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

2145 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)(3,4)(5,7)(6,8)(10,13)(11,15)(17,18)(19,23)(20,21)(22,24)(26,29)(27,31);(1,3,5,8)(2,6,7,4)(10,13)(14,16)(17,19,22,20)(18,21,24,23)(27,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,24,21)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,24,21)(25,27,28,31)(26,30,29,32);(1,4,5,6)(2,3,7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,23)(18,20,24,19)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=hi,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, 2853115);