#388460

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
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

1164 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)(7,17)(8,21,25,13)(9,22,27,35)(10,28)(11,16)(12,19)(14,30,33,26)(15,31)(18,29,34,24)(20,23)(32,36);(1,2)(3,4)(5,7,10,19,11,20,31,36)(6,12,15,17,16,32,28,23)(8,22,24,33,25,35,29,14)(9,21,26,34,27,13,30,18);(2,4)(5,8,10,29,11,25,31,24)(6,13,15,34,16,21,28,18)(7,14,19,35,20,33,36,22)(9,23,30,12,27,17,26,32);(5,9)(6,14)(7,18)(8,23)(10,30)(11,27)(12,29)(13,19)(15,35)(16,33)(17,25)(20,34)(21,36)(22,28)(24,32)(26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=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, 295656);