#212908

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

706 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120104);