#332596

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|b^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, 239792);