#4449657

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2845 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336784, #72336791, #72336857
maximal normal subgroup of	#72336784, #72336791, #72336857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4356853);