#333126

order	512
nilpotency class	3
isoclinism	isoclinic class of #333055
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	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,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, 240322);