#183614

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 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)(5,6,8,13,9,12,17,10)(7,14,16,18,15,11,19,20)(21,22,24,29,26,28,25,30)(23,31,32,35,33,36,34,27);(6,10)(8,17)(11,18)(12,13)(14,20)(16,19)(21,23,25,32,26,33,24,34)(22,27,30,31,28,35,29,36);(1,3,4,2)(5,7)(6,11)(8,16)(9,15)(10,18)(12,14)(13,20)(17,19)(22,28)(27,35)(29,30)(31,36);(6,12)(7,15)(10,13)(16,19)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,e^b=eh,f^b=fh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90810);