#116227

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
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

626 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)(8,11)(9,14)(10,13)(12,17)(16,19)(18,20)(21,22,27,30,25,31,26,36)(23,28,32,34,33,24,29,35);(6,11)(7,16,18,17)(12,15,19,20)(13,14)(21,23)(22,28)(24,31)(25,33)(26,29)(27,32)(30,34)(35,36);(2,4)(5,7,8,18)(6,12,11,19)(9,16,10,17)(13,20,14,15)(21,24,25,28)(22,29,31,32)(23,30,33,36)(26,35,27,34)

Polycyclic presentation:

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

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, 23423);