#117771

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

649 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,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, 24967);