#289109

order	512
nilpotency class	3
isoclinism	isoclinic class of #289075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,f^c=fi,g^c=ghi

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, 196305);