#391821

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=de,g^c=gi,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, 299017);