#8700347

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

3871 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,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, 8607543);