#234276

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1176 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:

(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(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);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,40)(26,38)(27,39)(28,35)(32,45)(33,47)(34,46)(37,42)(41,48)(43,44);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(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);(3,4)(8,16)(12,15)(13,14)(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=i,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,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, 141472);