#289144

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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)(9,10,14,20)(11,16,24,33)(12,25,28,36)(13,19,29,18)(15,23,31,22)(17,34,37,27)(21,38,26,30)(32,40,35,39);(1,3,6,2,4,8,5,7)(9,11,13,22,14,24,29,23)(10,15,18,16,20,31,19,33)(12,21,27,39,28,26,34,40)(17,30,25,32,37,38,36,35);(1,2,4,7)(3,5,8,6)(9,11)(10,16)(12,26)(13,22)(14,24)(15,19)(17,35)(18,31)(20,33)(21,28)(23,29)(25,30)(27,40)(32,37)(34,39)(36,38);(9,12,13,27,14,28,29,34)(10,17,18,25,20,37,19,36)(11,21,22,39,24,26,23,40)(15,30,16,32,31,38,33,35)

Polycyclic presentation:

2^9|a^2=fhi,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,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196340);