#234270

order	512
nilpotency class	3
isoclinism	isoclinic class of #234257
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1223 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)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,44)(31,38)(32,41)(33,43)(34,42)(36,39)(37,47)(40,45)(46,48);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,23,31)(18,24,27,39)(20,29,34,46)(21,30,22,36)(25,40,42,37)(26,38,35,28)(32,45,33,47)(41,48,43,44);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,23,34)(18,25,27,42)(19,29,31,46)(21,33,22,32)(24,37,39,40)(26,43,35,41)(28,44,38,48)(30,47,36,45);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,15,16,11)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,38,39,28)(25,41,42,43)(29,45,46,47)(37,44,40,48)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,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=di,e^b=eh,f^b=fi,d^c=dh,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, 141466);