#2503839

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

2626 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(25,26,29,31)(27,32,28,30);(1,2,4,6)(3,5,7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,22,23,24)(25,27,29,28)(26,30,31,32);(3,7)(5,8)(10,14)(13,15)(27,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,26)(27,32)(28,30)(29,31);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,24,23,22)(25,28,29,27)(26,30,31,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2411035);