#4869226

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

3847 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)(6,7)(9,10)(11,16)(12,14)(13,15)(18,21)(19,23)(26,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,14)(10,12,15,16)(18,21)(22,24)(25,26,29,30)(27,28,32,31);(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,30)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232585, #72232614, #72232659, …
maximal normal subgroup of	#72232585, #72232614, #72232659, …
maximal quotient of	#72328453, #72331306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4776422);