#6362703

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61434	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6269899);