#348836

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 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)(21,22,28,23)(24,33,25,36)(26,31,27,30)(29,35,32,34);(1,2,3,4)(5,7)(6,10)(8,15)(9,16)(11,18)(12,14)(13,19)(17,20)(21,23,28,22)(24,32,25,29)(26,31,27,30)(33,34,36,35);(6,11)(9,17)(10,18)(12,13)(14,19)(16,20)(21,24,28,25)(22,29,23,32)(26,35,27,34)(30,36,31,33);(7,15)(10,14)(16,20)(18,19)(21,25,28,24)(22,29,23,32)(26,34,27,35)(30,36,31,33)

Polycyclic presentation:

2^9|a^2=i,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, 256032);