#6350247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339705, #72339718, #72339721, …
maximal normal subgroup of	#72339705, #72339718, #72339721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257443);