#332828

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240024);