#240836

order	512
nilpotency class	3
isoclinism	isoclinic class of #240831
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

833 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:

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

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=di,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, 148032);