#323169

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
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

639 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=h,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230365);