#244259

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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)(3,6,7,8)(10,13)(14,16)(17,18,21,26,22,25,20,24)(19,27,29,23,30,31,28,32);(2,5)(6,8)(10,14)(12,15)(13,16)(17,19,22,30)(18,23,25,32)(20,29,21,28)(24,31,26,27);(9,10)(11,13)(12,16)(14,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,3,4,7)(2,6,5,8)(18,25)(23,32)(24,26)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151455);