#330973

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238169);