#113043

order	512
nilpotency class	3
isoclinism	isoclinic class of #96971
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di

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, 20239);