#116420

order	512
nilpotency class	3
isoclinism	isoclinic class of #116419
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5,6,7)(9,10,13,18,14,17,27,32)(11,15,21,30,20,29,36,39)(12,22,25,33,24,16,26,31)(19,34,35,40,23,28,37,38);(9,11)(10,15)(12,23,24,19)(13,21)(14,20)(16,28,22,34)(17,29)(18,30)(25,37,26,35)(27,36)(31,38,33,40)(32,39);(1,2)(3,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,22)(18,33)(20,23)(21,37)(25,27)(29,34)(30,40)(31,32)(35,36)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=ei,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, 23616);