#349709

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256905);