#6477592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470237
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 ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238047, #72251176, #72251177
maximal normal subgroup of	#72238047, #72251176, #72251177

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384788);