#233607

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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	#60942	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1178 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)(3,12)(4,14)(5,10)(6,16)(7,11)(8,13)(9,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,45)(30,38)(32,43)(33,40)(34,42)(35,41)(37,48)(39,46)(44,47);(1,3)(2,8)(4,11)(5,13)(6,14)(7,9)(10,15)(12,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,39)(26,43)(27,38)(28,36)(33,48)(34,47)(35,46)(37,42)(40,45)(41,44);(1,4,7,8)(2,9,11,3)(5,15,6,12)(10,13,16,14)(17,20,23,34)(18,25,27,42)(19,29,32,47)(21,35,22,33)(24,37,38,39)(26,41,36,40)(28,44,43,45)(30,48,31,46);(1,5,7,6)(2,10,11,16)(3,13,9,14)(4,12,8,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,40,42,41)(29,46,47,48)(37,45,39,44)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,d^c=dh,e^c=ei,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, 140803);