#115565

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22761);