#2798240

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

2242 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)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(19,23)(22,24)(25,26)(27,32)(28,30)(29,31);(1,2,5,3)(4,6,8,7)(9,11,12,16)(10,13,15,14)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,31,28,32);(2,3)(6,7)(11,16)(13,14)(26,28)(31,32);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,32,29,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,31,28,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2705436);