#243999

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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)(9,10,12,14,13,15,11,16)(17,18,20,25,21,24,30,23)(19,26,28,31,29,32,27,22);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,27)(21,29)(23,26)(24,31)(25,32)(28,30);(9,10)(11,14)(12,16)(13,15)(18,23)(19,27)(20,30)(24,25)(26,32)(28,29);(9,11,13,12)(10,14,15,16)(18,24)(22,31)(23,25)(26,32)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=hi,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, 151195);