#330981

order	512
nilpotency class	3
isoclinism	isoclinic class of #330975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

512 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)(3,6)(4,8)(5,7)(9,13)(10,15)(11,14)(12,16)(18,23)(19,27)(20,31)(21,33)(24,37)(28,39)(29,43)(30,34)(38,48)(40,47)(41,46)(42,44);(1,3)(2,6)(4,9)(5,11)(7,14)(8,13)(10,12)(15,16)(17,18,21,26,22,23,33,36)(19,28,27,37,30,24,34,39)(20,25,32,41,31,35,45,46)(29,40,42,47,44,38,43,48);(3,9)(4,12)(6,13)(8,16)(10,11)(14,15)(17,19,22,30)(18,24,23,28)(20,29,31,44)(21,34,33,27)(25,40,35,38)(26,37,36,39)(32,43,45,42)(41,48,46,47);(2,7)(6,14)(8,16)(13,15)(17,20,22,31)(18,25,23,35)(19,29,30,44)(21,32,33,45)(24,38,28,40)(26,41,36,46)(27,42,34,43)(37,47,39,48)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238177);