#290071

order	512
nilpotency class	3
isoclinism	isoclinic class of #290011
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

614 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)(17,28)(18,21)(22,38)(24,31)(25,33)(27,30)(29,35)(32,39)(34,40)(36,37);(1,3,4,2,5,8,6,7)(9,11,21,24,14,25,13,23)(10,15,19,33,20,31,18,16)(12,22,37,40,29,27,28,39)(17,30,35,34,36,38,26,32);(1,2,5,7)(3,6,8,4)(9,11,14,25)(10,16,20,33)(12,27,29,22)(13,23,21,24)(15,18,31,19)(17,34,36,32)(26,30,35,38)(28,40,37,39);(1,4,5,6)(2,3,7,8)(9,12,13,28,14,29,21,37)(10,17,18,26,20,36,19,35)(11,22,23,39,25,27,24,40)(15,30,16,32,31,38,33,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=hi,d^2=e,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,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, 197267);