#2600325

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2418 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=hi,d^2=fi,e^2=fhi,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, 2507521);