#323530

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

686 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egh,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,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, 230726);