#290008

order	512
nilpotency class	3
isoclinism	isoclinic class of #289947
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×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=hi,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,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, 197204);