#213198

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

714 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)(6,10)(8,16)(11,17)(12,18)(13,20)(15,19)(21,22)(23,31)(24,26)(25,29)(27,30)(28,33)(32,35)(34,36);(5,6,8,10,9,12,16,18)(7,13,15,20,14,11,19,17)(21,23,26,34,27,35,25,33)(22,28,29,32,30,36,24,31);(5,7)(6,11)(8,15)(9,14)(10,17)(12,13)(16,19)(18,20)(21,24,25,30,27,29,26,22)(23,32,33,28,35,31,34,36);(1,3,4,2)(6,12)(7,14)(10,18)(15,19)(21,25,27,26)(22,29,30,24)(23,33,35,34)(28,32,36,31)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,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, 120394);