#2580675

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1901 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=hi,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, 2487871);