#229721

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=di,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136917);