#232798

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1214 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)(3,11)(4,7)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,31)(25,36)(29,46)(30,40)(32,41)(34,45)(35,43)(37,44)(39,47)(42,48);(2,7)(3,12)(4,6)(9,10)(13,14)(15,16)(17,19)(18,24)(20,34)(21,32)(22,31)(23,30)(25,42)(26,41)(27,40)(28,38)(29,35)(33,39)(36,47)(37,48)(43,45)(44,46);(1,3,5,14)(2,8,9,11)(4,12,6,13)(7,15,10,16)(17,20,23,37)(18,25,26,43)(19,29,30,47)(21,35,22,36)(24,39,41,46)(27,44,38,33)(28,45,40,42)(31,34,32,48);(2,9)(7,10)(8,11)(15,16)(18,26)(19,30)(25,43)(27,38)(29,47)(31,32)(33,44)(34,48)

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

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, 139994);