#119619

order	512
nilpotency class	3
isoclinism	isoclinic class of #119611
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26815);