#290057

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

580 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=hi,d^2=ei,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, 197253);