#234450

order	512
nilpotency class	3
isoclinism	isoclinic class of #234438
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	C₂³×C₄	C₂⁴
prime radical	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1283 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)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,37)(28,35)(29,36)(30,38)(31,39)(32,40)(33,41)(34,43)(42,47)(44,46)(45,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,37)(25,35)(26,36)(30,42)(31,43)(32,44)(33,45)(34,39)(38,47)(40,46)(41,48);(4,5)(7,15)(8,9)(12,14)(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,32,33,30)(23,34,36,37)(25,40,41,38)(28,44,45,42)(35,47,48,46);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,21,22,33)(18,25,26,41)(19,28,29,45)(20,30,31,32)(23,35,36,48)(24,38,39,40)(27,42,43,44)(34,46,37,47)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141646);