#6571612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533135
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 ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268363, #72268399
maximal normal subgroup of	#72268363, #72268399

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478808);