#304632

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

832 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)(11,15)(12,16)(13,14)(17,18)(19,28)(20,30)(21,27)(22,26)(23,25)(24,29)(31,32);(1,3,5,4)(2,6,8,7)(10,14)(11,16)(17,19)(18,24)(20,21)(22,29)(23,31)(25,30)(26,28)(27,32);(1,3,5,4)(2,7,8,6)(9,11,15,10,13,16,12,14)(17,20,21,19,22,31,23,29)(18,24,25,32,26,28,27,30);(1,4,5,3)(2,7,8,6)(9,12,13,15)(10,11,14,16)(17,21,22,23)(18,25,26,27)(19,20,29,31)(24,30,28,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=ef,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, 211828);