#213139

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

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

790 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)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(22,26)(23,29)(24,32)(27,30)(28,34)(31,35);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20)(21,22,24,26,25,28,32,34)(23,30,29,36,31,27,35,33);(7,14)(10,19)(13,20)(15,18)(21,23,24,29,25,31,32,35)(22,27,26,33,28,30,34,36);(1,3,4,2)(22,28)(26,34)(27,30)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di

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, 120335);