#2549036

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2659 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)(3,8)(4,6)(5,7)(9,10,14,16)(11,12,13,15)(18,19)(20,24)(26,28)(30,32);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,15,16,12)(18,19)(22,23)(27,31)(30,32);(3,4)(6,8)(10,16)(11,13)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,31,32);(1,2)(3,8)(4,6)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,32)(29,31);(1,4,5,3)(2,6,7,8)(9,13,14,11)(10,12,16,15)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=fghi,d^2=fh,e^2=ghi,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, 2456232);