#331978

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	#3207	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,11)(3,13,9,8)(4,15,10,14)(6,16,7,12)(17,18)(19,28)(20,25)(21,33)(22,36)(23,27)(24,29)(26,32)(30,47)(31,41)(34,43)(35,42)(37,40)(38,44)(39,48)(45,46);(1,3)(2,8)(4,12)(5,9)(6,14)(7,15)(10,16)(11,13)(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);(1,4,5,10)(2,9,11,3)(6,8,7,13)(12,14,16,15)(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);(1,3,5,9)(2,10,11,4)(6,15,7,14)(8,12,13,16)(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=i,c^2=hi,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, 239174);