#6357651

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62417	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1862 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264847);