#6357309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357300
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1946 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:

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

Polycyclic presentation:

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

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, 6264505);