#2942489

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2746 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)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,25,30)(20,27,28,24)(26,32,29,31);(2,4)(6,8)(10,13)(14,15)(18,22)(19,25)(20,28)(31,32);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,28)(29,32);(2,4)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,28)(27,31);(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,20,21,28)(18,24,22,27)(19,26,25,29)(23,31,30,32)

Polycyclic presentation:

2^9|a^2=g,c^2=fhi,d^2=fgi,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, 2849685);