#333261

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#560
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,f^a=fi,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, 240457);