#226121

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, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fhi,d^2=hi,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, 133317);