#1452038

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2201 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gi,c^2=f,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1359234);