#8700449

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607645);