#240136

order	512
nilpotency class	3
isoclinism	isoclinic class of #240131
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

795 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:

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

Polycyclic presentation:

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

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, 147332);