#620121

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

2138 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,16,12,15)(17,18)(19,20)(21,22)(23,24)(26,29)(27,31);(1,2,4,6)(3,5,8,7)(10,14)(15,16)(18,22)(19,24)(26,29)(30,32);(9,11,13,12)(10,15,14,16)(17,19,21,24)(18,23,22,20)(25,26,28,29)(27,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,16,14,15)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 527317);