#6477585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470225
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285303, #72285362, #72317879
maximal normal subgroup of	#72285303, #72285362, #72317879

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384781);