#1118210

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2290 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:

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

Polycyclic presentation:

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

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, 1025406);