#232519

order	512
nilpotency class	3
isoclinism	isoclinic class of #232515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37446	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139715);