#274058

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

813 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,d^2=hi,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, 181254);