#332877

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240073);