#326554

order	512
nilpotency class	3
isoclinism	isoclinic class of #251791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2580
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233750);