#240233

order	512
nilpotency class	3
isoclinism	isoclinic class of #240223
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 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)(3,7,12,16)(4,9,5,8)(11,14,13,15)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,30)(25,35)(29,44)(31,40)(32,43)(34,39)(36,42)(37,47)(41,45)(46,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,23,32)(18,24,27,40)(20,34,35,46)(21,31,22,30)(25,41,33,44)(26,43,38,28)(29,36,45,37)(39,47,48,42);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20,21,36,23,35,22,37)(18,25,26,42,27,33,38,47)(19,29,30,46,32,45,31,34)(24,39,28,41,40,48,43,44);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(20,35)(25,33)(29,45)(34,46)(36,37)(39,48)(41,44)(42,47)

Polycyclic presentation:

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

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, 147429);