#6347102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347092
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

3553 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)(3,12)(4,9)(5,18)(6,11)(7,10)(8,13)(14,26)(15,21)(16,24)(17,25)(19,23)(20,29)(22,28)(27,32)(30,31);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,21)(10,22)(11,23)(12,18)(16,27)(17,28)(20,30)(24,31)(25,32)(26,29);(1,4,16,20)(2,9,24,29)(3,13,27,30)(5,7,17,6)(8,21,31,26)(10,25,11,18)(12,22,32,23)(14,28,15,19);(2,10)(8,22)(9,25)(11,24)(12,26)(18,29)(21,32)(23,31);(2,11)(8,23)(9,18)(10,24)(12,21)(22,31)(25,29)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240205, #72284955
maximal normal subgroup of	#72240205, #72284955
maximal quotient of	#72240405

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254298);