#332880

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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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)(17,18,22,24,23,28,21,27)(19,25,36,37,31,40,29,39)(20,32,35,26,34,42,33,41)(30,44,43,38,46,47,45,48);(1,3,5,6)(2,7,10,9)(4,11,13,14)(8,12,16,15)(18,24)(19,29)(20,33)(21,22)(26,41)(27,28)(31,36)(34,35)(37,39)(38,44)(43,45)(47,48);(1,4,5,13)(2,8,10,16)(3,11,6,14)(7,12,9,15)(17,19,23,31)(18,25,28,40)(20,30,34,46)(21,29,22,36)(24,37,27,39)(26,38,41,48)(32,44,42,47)(33,45,35,43);(2,9)(3,6)(7,10)(8,15)(11,14)(12,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,32)(25,38)(27,42)(28,41)(29,43)(31,46)(36,45)(37,44)(39,47)(40,48)

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=hi,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, 240076);