#240850

order	512
nilpotency class	3
isoclinism	isoclinic class of #240845
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,e^2=h,f^2=i,g^2=h,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,g^b=gh,d^c=dhi,f^c=fhi

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, 148046);