#343124

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18687	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 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:

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

Polycyclic presentation:

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

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, 250320);