#1316556

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2196 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1223752);