#6190485

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

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

3396 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,21)(19,22,24,23)(25,26)(27,31)(28,30)(29,32);(3,7)(6,8)(10,14)(12,15)(17,18,20,21)(19,23,24,22)(25,27,28,32)(26,29,30,31);(2,5)(6,8)(10,14)(12,15)(18,21)(22,23)(26,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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	#72300841, #72300853, #72300901, …
maximal normal subgroup of	#72300841, #72300853, #72300901, …
maximal quotient of	#72328878, #72330350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6097681);