#391222

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1097 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298418);