#2242186

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1860 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,15)(13,14)(18,19)(20,24)(25,26,29,28)(27,30,32,31);(2,3)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18,21,19)(20,22,24,23)(25,27,29,32)(26,30,28,31);(4,8)(6,7)(10,13)(11,15)(17,18,21,19)(20,23,24,22)(25,27)(26,30)(28,31)(29,32);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,15,13,11)(17,19,21,18)(20,23,24,22);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,15,12)(17,20)(18,22)(19,23)(21,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2149382);