#6357674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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	#60817	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1915 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh,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, 6264870);