#8745966

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,9)(4,8)(5,10)(6,11)(7,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,32)(22,27)(23,30)(24,31)(26,29)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19)(18,23)(20,21)(22,30)(24,32)(25,26)(27,28)(29,31)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,4,6,10)(2,8,11,5)(3,12,14,16)(7,15,9,13)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,32,28,26)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,4)(2,8)(3,12)(5,11)(6,10)(7,15)(9,13)(14,16)(17,20)(18,25)(19,29)(21,30)(22,31)(23,32)(24,27)(26,28)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,3)(2,9)(4,12)(5,15)(6,14)(7,11)(8,13)(10,16)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,25,28,24);(1,5,6,8)(2,10,11,4)(3,13,14,15)(7,16,9,12)(17,18)(19,23)(20,24)(21,32)(22,27)(25,31)(26,29)(28,30)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653162);