#6347429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347429
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3559 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)(3,11)(4,15)(5,9)(6,8)(7,13)(10,18)(12,24)(14,25)(16,21)(17,28)(19,22)(20,27)(23,30)(26,32)(29,31);(1,3)(2,7)(4,16)(5,12)(6,14)(8,20)(9,21)(10,24)(11,15)(13,18)(17,29)(19,26)(22,31)(23,32)(25,28)(27,30);(4,17)(10,23)(11,25)(13,27)(15,28)(16,29)(18,30)(24,32);(2,8)(7,20)(9,22)(10,23)(11,25)(15,28)(21,31)(24,32);(2,9)(3,12)(8,22)(10,15)(13,16)(14,26)(23,28)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284955
maximal normal subgroup of	#72284955

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254625);