#232441

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	#9974	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1268 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,30)(25,36)(29,46)(31,41)(32,40)(34,45)(35,44)(37,43)(39,47)(42,48);(6,12)(13,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,39)(36,47)(37,48)(43,46)(44,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,39)(26,43)(27,44)(28,45)(30,34)(31,48)(32,47)(33,38)(40,42)(41,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139637);