#115889

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

620 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)(7,13)(8,16)(9,12)(10,14)(11,17)(15,20)(18,19)(21,22,25,28,26,30,24,29)(23,27,31,34,33,35,32,36);(1,2,3,4)(8,17)(10,19)(11,16)(14,18)(21,23)(22,27)(24,32)(25,31)(26,33)(28,34)(29,36)(30,35);(5,7,9,15)(6,10,12,19)(8,18,17,14)(11,20,16,13)(22,28)(23,31)(24,25)(29,30)(32,33)(34,36)

Polycyclic presentation:

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

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, 23085);