#6477583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470225
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3580 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:

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

Polycyclic presentation:

2^9|c^2=h,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	#72278134, #72278163, #72278200, …
maximal normal subgroup of	#72278134, #72278163, #72278200, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384779);