#8745860

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

4098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=ghi,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, 8653056);