#119341

order	512
nilpotency class	3
isoclinism	isoclinic class of #119339
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

564 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)(9,10,13,18,14,17,27,32)(11,19,21,33,22,15,25,29)(12,16,24,31,23,30,26,34)(20,35,37,40,36,28,38,39);(1,3,6,2)(4,5,8,7)(9,11,14,22)(10,15,17,19)(12,20,23,36)(13,25,27,21)(16,28,30,35)(18,33,32,29)(24,38,26,37)(31,40,34,39);(1,4,5,2,6,8,7,3)(9,12)(10,16,17,30)(11,20)(13,26)(14,23)(15,28,19,35)(18,34,32,31)(21,38)(22,36)(24,27)(25,37)(29,40,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=i,g^2=h,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=fi,g^b=ghi,e^c=eh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26537);