#640956

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2654 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 548152);