#333208

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eh,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,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, 240404);