#115824

order	512
nilpotency class	3
isoclinism	isoclinic class of #100027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,g^c=gh

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, 23020);