#226088

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fhi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133284);