#696047

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2082 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,d^2=fg,e^2=g,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, 603243);