#2433678

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2254 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=hi,d^2=hi,e^2=gh,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, 2340874);