#360285

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1843 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)(7,12,8,19)(9,14,23,29)(10,16)(11,15)(13,22,27,18)(17,25)(20,30)(21,32)(24,28,33,36)(26,31)(34,35);(1,2)(3,4)(5,7,11,18,10,8,20,22)(6,12,15,13,16,19,30,27)(9,17,24,31,23,21,33,34)(14,25,28,26,29,32,36,35);(2,4)(5,8)(6,13)(7,10)(9,21)(11,18)(12,15)(14,26)(16,27)(17,23)(19,30)(20,22)(24,31)(25,28)(29,35)(32,36)(33,34);(5,9,10,23)(6,14,16,29)(7,17,8,21)(11,24,20,33)(12,25,19,32)(13,26,27,35)(15,28,30,36)(18,31,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267481);