#119155

order	512
nilpotency class	3
isoclinism	isoclinic class of #119152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26351);