#115563

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

908 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)(5,6,9,14,10,11,8,13)(7,15,17,12,18,20,16,19)(21,22,25,31)(23,27)(24,32)(26,29,35,30)(28,36)(33,34);(2,4)(6,11)(8,9)(15,20)(16,17)(21,23)(22,27)(24,31)(25,32)(26,28)(29,36)(30,34)(33,35);(5,7,10,18)(6,12,11,19)(8,17,9,16)(13,20,14,15)(21,24,25,27)(22,28,31,33)(23,30,32,29)(26,36,35,34)

Polycyclic presentation:

2^9|a^2=g,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, 22759);