#705533

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2310 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(18,22)(19,23)(27,31)(29,32);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,12,14,15)(17,18)(19,20)(21,22)(23,24);(2,3)(6,7)(10,14)(12,15)(18,22)(19,23)(25,26,28,30)(27,29,31,32);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,19,21,23)(18,20,22,24)(25,27)(26,29)(28,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,23)(19,22)(21,24)

Polycyclic presentation:

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

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, 612729);