#333146

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
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	#18696	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240342);