#6357919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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	#62423	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1939 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,28)(20,29)(21,26)(22,32)(23,27)(24,25)(30,31);(2,6)(7,14)(8,13)(9,15)(17,19,21,30,23,29,22,25)(18,24,32,20,27,31,26,28);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,21,31,23,28,22,24)(18,25,32,19,27,30,26,29);(3,10)(7,14)(9,15)(11,16)(17,18)(19,24)(20,30)(21,32)(22,26)(23,27)(25,28)(29,31);(17,21,23,22)(18,26,27,32)(19,25,29,30)(20,31,28,24)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di,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, 6265115);