#321913

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
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

661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229109);