#119509

order	512
nilpotency class	3
isoclinism	isoclinic class of #119502
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

621 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:

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

Polycyclic presentation:

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

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, 26705);