#331944

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18856	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,23,27)(19,28,31,41)(20,25,22,36)(21,33,35,42)(24,37,40,29)(26,38,44,32)(30,47,48,39)(34,43,46,45);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19)(18,24)(20,29)(21,34)(22,37)(23,31)(25,41)(26,43)(27,40)(28,36)(30,32)(33,39)(35,46)(38,48)(42,47)(44,45);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,32,35,38)(24,28,40,41)(26,42,44,33)(30,46,48,34)(39,45,47,43);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,11,9,15)(17,21,23,35)(18,26,27,44)(19,30,31,48)(20,32,22,38)(24,39,40,47)(25,42,36,33)(28,45,41,43)(29,46,37,34)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239140);