#2575876

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2029 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fh,d^2=ghi,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, 2483072);