#6629051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629051
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4428 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215707, #72215784, #72215931, …
maximal normal subgroup of	#72215784, #72215931, #72219691, …
maximal quotient of	#72312265, C₂×#6629051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536247);