#332834

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

573 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dei,e^b=eh,f^b=fh,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, 240030);