#8346005

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

4296 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253201);