#237720

order	512
nilpotency class	3
isoclinism	isoclinic class of #237707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1264 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,24)(21,33)(22,26)(23,29)(25,36)(28,44)(30,42)(31,40)(32,43)(34,38)(35,41)(37,47)(39,46)(45,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,22,30)(18,23,26,38)(20,28,31,45)(21,34,36,29)(24,39,40,37)(25,42,33,27)(32,47,35,46)(41,48,43,44);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20)(18,24)(19,28)(21,35)(22,31)(23,37)(25,43)(26,40)(27,44)(29,47)(30,45)(32,36)(33,41)(34,46)(38,39)(42,48);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(20,31)(24,40)(28,45)(32,35)(37,39)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=h,d^2=i,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,d^c=dhi,e^c=eh,f^c=fi

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, 144916);