#304791

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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×#459	C₂²
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)(11,16)(13,15)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,22,26,29)(20,24,27,30)(25,28,31,32);(1,3,5,4)(2,7,8,6)(10,13,14,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,5,3)(2,7,8,6)(18,23)(19,26)(24,30)(25,31)

Polycyclic presentation:

2^9|a^2=e,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,d^b=dh,f^b=fi,g^b=gi,d^c=dh

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, 211987);