#6350852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2847 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258048);