#240630

order	512
nilpotency class	3
isoclinism	isoclinic class of #240625
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,22,26)(19,27,29,42)(20,31,33,24)(21,35,37,25)(23,36,39,30)(28,44,32,43)(34,41,48,46)(38,45,40,47);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,19)(18,23)(20,32)(21,36)(22,29)(24,40)(25,42)(26,39)(27,35)(28,33)(30,37)(31,38)(34,45)(41,43)(44,46)(47,48);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,20,21,34,22,33,37,48)(18,24,25,41,26,31,35,46)(19,28,30,45,29,32,36,47)(23,38,27,43,39,40,42,44);(19,29)(20,33)(23,39)(24,31)(27,42)(30,36)(34,48)(41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147826);