#851162

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 758358);