#234213

order	512
nilpotency class	3
isoclinism	isoclinic class of #234178
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1172 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)(3,7)(4,9)(5,13)(6,10)(8,11)(12,16)(14,15)(17,18,23,27)(19,28,31,38)(20,25,34,42)(21,26,22,35)(24,36,39,30)(29,44,46,48)(32,41,33,43)(37,45,40,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(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);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(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)(2,9,10,13)(3,11,12,14)(7,8,16,15)(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|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fi

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, 141409);