#252267

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#9914	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1333 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,15)(14,16)(17,19,22,18)(20,29,31,28)(21,27,23,25)(24,30,26,32);(10,13)(11,16)(14,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,29)(27,28);(2,6)(5,7)(17,21,22,23)(18,25,19,27)(20,30,31,32)(24,29,26,28)

Polycyclic presentation:

2^9|b^2=e,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159463);