#290000

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	#28174	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,8,5,10,4,9)(3,7,11,16,12,14,13,15)(17,18)(19,24)(20,34)(21,27)(22,28)(23,31)(25,35)(26,29)(30,44)(32,38)(33,40)(36,41)(37,47)(39,45)(42,43)(46,48);(1,3,4,13,5,12,6,11)(2,7,9,15,10,14,8,16)(17,19,29,32,22,33,21,31)(18,23,27,40,28,38,26,24)(20,30,43,46,36,47,35,45)(25,37,41,48,42,44,34,39);(2,8)(3,11)(4,6)(7,14)(9,10)(12,13)(17,19,22,33)(18,24,28,40)(20,30,36,47)(21,31,29,32)(23,26,38,27)(25,39,42,48)(34,44,41,37)(35,45,43,46);(3,12)(7,14)(11,13)(15,16)(17,20,29,43,22,36,21,35)(18,25,27,41,28,42,26,34)(19,30,32,46,33,47,31,45)(23,37,40,48,38,44,24,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=h,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, 197196);