#8407120

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

4374 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314316);