#259065

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166261);