#4320035

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334255
maximal normal subgroup of	#72334255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4227231);