#348810

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,14)(8,12)(9,11)(10,15)(13,17)(16,19)(18,20)(23,30)(24,31)(25,33)(27,34)(28,35)(32,36);(1,2,3,4)(5,7)(6,10)(8,15)(9,16)(11,18)(12,14)(13,19)(17,20)(21,22,26,29)(23,27,24,32)(25,28,33,35)(30,34,31,36);(6,11)(9,17)(10,18)(12,13)(14,19)(16,20)(21,23,26,24)(22,27,29,32)(25,31,33,30)(28,36,35,34);(7,15)(10,14)(16,20)(18,19)(21,24,26,23)(22,27,29,32)(25,30,33,31)(28,36,35,34)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256006);