#1438098

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2112 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)(4,8)(5,6)(10,14)(11,16)(18,21)(22,24)(26,29)(27,31);(2,6)(4,7)(9,10,13,14)(11,15,16,12)(26,29)(30,32);(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19,20,23)(18,22,21,24);(1,4,5,7)(2,8,6,3)(9,12,13,15)(10,16,14,11)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=f,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, 1345294);