#615478

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
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

2012 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=f,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522674);