#333153

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	#18687	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

631 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=h,d^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, 240349);