#380184

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287380);