#317660

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

618 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:

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

Polycyclic presentation:

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

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, 224856);