#4474758

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3751 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:

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300937, #72329055, #72329079, …
maximal normal subgroup of	#72300937, #72329055, #72329079, …
maximal quotient of	#72330850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4381954);