#115589

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fi,d^c=di,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22785);