#876983

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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

2374 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)(10,12)(14,15)(17,18,22,24)(19,23,21,20)(27,31)(30,32);(1,2,6,5)(3,7,4,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,20,24,23)(26,29)(30,32);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,14,16,15)(17,20,22,23)(18,19,24,21)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,8,5,7)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,23)(22,24)(25,27,28,31)(26,30,29,32);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,23,24,20)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=f,d^2=fi,e^2=f,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, 784179);