#6164392

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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

2754 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326558, #72326573, #72326641
maximal normal subgroup of	#72326558, #72326573, #72326641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6071588);