#234193

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

1243 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(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);(4,13)(6,15)(7,12)(10,14)(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);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(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);(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,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, 141389);