#234266

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

1188 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)(17,18,22,26)(19,27,30,42)(20,24,31,40)(21,33,36,25)(23,34,38,29)(28,44,45,48)(32,43,35,41)(37,46,39,47);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19,22,30)(18,23,26,38)(20,28,31,45)(21,34,36,29)(24,39,40,37)(25,42,33,27)(32,47,35,46)(41,48,43,44);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(17,20)(18,24)(19,28)(21,35)(22,31)(23,37)(25,43)(26,40)(27,44)(29,47)(30,45)(32,36)(33,41)(34,46)(38,39)(42,48);(3,4)(8,16)(12,15)(13,14)(20,31)(24,40)(28,45)(32,35)(37,39)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,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, 141462);