#2555258

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

1846 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fghi,d^2=f,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, 2462454);