#6529158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499870
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

3564 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)(7,19)(8,21)(11,25)(12,26)(16,28)(17,29)(23,32);(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,22)(13,25)(15,21)(16,20)(17,23)(18,24)(19,27)(26,31)(28,30)(29,32);(1,3,5,14)(2,7,9,11)(4,16,17,12)(6,13,18,27)(8,22,23,20)(10,19,24,25)(15,28,29,26)(21,31,32,30);(1,4,5,17)(2,8,9,23)(3,12,14,16)(6,15,18,29)(7,20,11,22)(10,21,24,32)(13,26,27,28)(19,30,25,31);(2,9)(7,11)(8,23)(10,24)(19,25)(20,22)(21,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436354);