#274005

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181201);