#243927

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,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, 151123);