#8418437

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	#42608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,e^2=g,f^2=gi,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, 8325633);