#380183

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,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, 287379);