#304635

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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)(9,10,13,14)(11,12,15,16)(17,18,22,27)(19,25,29,24)(20,31,30,32)(21,28,23,26);(1,3,5,4)(2,6,8,7)(10,14)(11,15)(17,19)(18,24)(20,21)(22,29)(23,30)(25,27)(26,31)(28,32);(1,3,5,4)(2,7,8,6)(9,11,12,14,13,15,16,10)(17,20,21,19,22,30,23,29)(18,25,26,31,27,24,28,32);(1,4,5,3)(2,7,8,6)(9,12,13,16)(10,11,14,15)(17,21,22,23)(18,26,27,28)(19,20,29,30)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211831);