#851165

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2052 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=fg,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, 758361);