#2832381

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=fgh,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, 2739577);