#8720518

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

3937 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,23,24)(19,22,20,25)(21,29,31,26)(27,30,28,32)(34,37)(35,36)(42,45)(43,44);(2,7)(3,12)(9,14)(10,13)(17,19)(18,22)(20,23)(21,27)(24,25)(26,32)(28,31)(29,30)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(3,12)(9,14)(10,13)(18,24)(19,20)(26,29)(27,28)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,30,29,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,32,25,30)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22,23,25)(18,20,24,19)(21,30,31,32)(26,27,29,28)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627714);