#698345

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=gh,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, 605541);