#117774

order	512
nilpotency class	3
isoclinism	isoclinic class of #117771
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

603 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,d^c=di,f^c=fi,g^c=gh

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, 24970);