#332746

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

588 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)(3,10)(4,6)(5,7)(8,14)(9,13)(11,16)(12,15)(17,18,23,28)(19,29,33,40)(20,34,37,26)(21,24,22,27)(25,30,41,32)(31,43,45,48)(35,42,36,38)(39,44,47,46);(2,6)(7,9)(8,15)(10,16)(18,24)(19,30)(20,35)(21,22)(27,28)(29,40)(31,45)(32,33)(36,37)(38,42)(39,48)(43,47);(2,7)(5,13)(6,9)(8,16)(10,15)(12,14)(17,19)(18,25)(20,31)(21,30)(22,32)(23,33)(24,29)(26,39)(27,40)(28,41)(34,47)(35,44)(36,46)(37,45)(38,43)(42,48);(1,3,4,11)(2,8,6,15)(5,14,13,12)(7,16,9,10)(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,38)(25,39)(27,42)(28,34)(29,43)(30,44)(32,46)(33,45)(40,48)(41,47)

Polycyclic presentation:

2^9|a^2=h,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, 239942);