#238978

order	512
nilpotency class	3
isoclinism	isoclinic class of #238970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,31)(25,36)(29,44)(30,41)(32,40)(34,39)(35,43)(37,47)(42,46)(45,48);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,40)(27,41)(28,38)(29,37)(33,44)(35,46)(36,45)(39,43)(47,48);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,35,23,36,22,37)(18,25,26,43,27,33,38,47)(19,29,31,45,32,46,30,34)(24,39,28,42,41,48,40,44);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,36,37)(24,40,41,28)(25,43,33,47)(29,34,46,45)(39,44,48,42)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=di,f^c=fhi,g^c=gi

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, 146174);