#1452047

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

2265 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=f,d^2=fgh,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, 1359243);