#242375

order	512
nilpotency class	3
isoclinism	isoclinic class of #242365
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,16)(14,15)(17,18,21,27,23,28,22,26)(19,29,31,40,33,39,32,24)(20,25,37,42,36,43,35,44)(30,45,47,38,46,41,34,48);(1,3,4,13,6,12,5,14)(2,7,8,11,9,15,10,16)(17,19,22,32,23,33,21,31)(18,24,27,29,28,40,26,39)(20,34,35,46,36,47,37,30)(25,41,42,48,43,45,44,38);(2,8)(3,12)(4,5)(7,11)(9,10)(15,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,44)(28,43)(29,45)(31,34)(32,47)(33,46)(39,41)(40,48);(2,9)(7,15)(8,10)(11,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,43,44)(30,34,46,47)(38,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149571);