#114128

order	512
nilpotency class	3
isoclinism	isoclinic class of #114125
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

848 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)(11,19,22,15)(12,16)(13,18)(14,17)(20,37,38,29)(21,35,26,30)(23,40,39,31)(24,33)(25,32)(27,36)(28,34);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,23,25,39)(13,26,28,21)(16,31,32,40)(18,35,34,30)(20,27,38,24)(29,36,37,33);(1,2)(3,7)(4,6)(5,8)(9,12,14,25)(10,16,17,32)(11,20,22,38)(13,27,28,24)(15,29,19,37)(18,36,34,33)(21,39,26,23)(30,40,35,31)

Polycyclic presentation:

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

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, 21324);