#234256

order	512
nilpotency class	3
isoclinism	isoclinic class of #234235
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

1191 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)(3,12)(4,15)(5,9)(6,10)(7,11)(8,14)(13,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,44)(29,42)(31,43)(32,41)(34,38)(35,40)(37,47)(39,46)(45,48);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,12,10,15)(17,19)(18,23)(20,28)(21,34)(22,29)(24,39)(25,42)(26,38)(27,33)(30,36)(31,47)(32,45)(35,46)(37,41)(40,48)(43,44);(1,4,7,3)(2,8,11,16)(5,14,6,13)(9,12,10,15)(17,20,22,32)(18,24,26,41)(19,28,29,45)(21,35,36,31)(23,37,38,39)(25,43,33,40)(27,44,42,48)(30,47,34,46);(1,5,7,6)(2,9,11,10)(3,13,4,14)(8,12,16,15)(19,29)(23,38)(27,42)(28,45)(30,34)(37,39)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^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=dhi,e^b=eh,f^b=fi,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141452);