#355555

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262751);