#332620

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
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

595 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)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,23,28)(19,29,32,37)(20,26,35,41)(21,24,22,27)(25,36,40,30)(31,45,47,43)(33,42,34,38)(39,44,48,46);(18,24)(19,30)(20,33)(21,22)(26,41)(27,28)(29,37)(32,36)(34,35)(39,45)(43,48)(44,46);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,19,23,32)(18,25,28,40)(20,31,35,47)(21,30,22,36)(24,37,27,29)(26,39,41,48)(33,46,34,44)(38,45,42,43);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,20,23,35)(18,26,28,41)(19,31,32,47)(21,34,22,33)(24,38,27,42)(25,39,40,48)(29,43,37,45)(30,44,36,46)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,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=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239816);