#256620

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59212	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,28)(20,30)(21,27)(22,26)(23,25)(24,29)(31,32);(1,3,5,4)(2,6,7,8)(9,11,14,15)(10,12,16,13)(17,19)(18,24)(20,21)(22,29)(23,31)(25,30)(26,28)(27,32);(1,4,5,3)(2,6,7,8)(10,11)(12,13)(15,16)(17,20,22,31)(18,24,26,28)(19,21,29,23)(25,30,27,32);(2,7)(6,8)(17,21,22,23)(18,25,26,27)(19,20,29,31)(24,30,28,32)

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=hi,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,f^c=fh,g^c=gi

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, 163816);