#6346760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346727
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307970, C₂×#6346760
maximal normal subgroup of	#72307970, C₂×#6346760
maximal quotient of	C₂×#6346760, #72309292, #72309293, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253956);