#6353457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
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
prime radical	#63014	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2972 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,23,18)(20,26,30,24)(21,27,22,25)(28,31,32,29);(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=hi,e^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,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260653);