#751607

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

2036 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)(6,8)(10,13)(14,15)(17,18,21,24)(19,22,20,23)(25,26,30,29)(27,32,28,31);(9,10)(11,15)(12,13)(14,16)(17,19,21,20)(18,22,24,23)(25,27,30,28)(26,31,29,32);(9,10,12,13)(11,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,26,30,29)(27,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,27)(26,31,29,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20,21,19)(18,23,24,22)(25,29,30,26)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=fh,e^2=fhi,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, 658803);