#880595

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2360 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=ghi,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, 787791);