#2594782

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1828 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)(3,7)(4,6)(5,8)(11,15)(14,16)(19,20)(22,23)(25,26,29,32)(27,31,28,30);(1,3,6,5)(2,7,4,8)(17,18,21,24)(19,22,20,23)(25,27,29,28)(26,30,32,31);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,26)(27,31)(28,30)(29,32);(1,4,6,2)(3,7,5,8)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,23)(21,24);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=i,e^2=fghi,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, 2501978);