#8406850

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	#52427	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,6)(3,9)(8,13)(11,15)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(34,38)(35,42)(40,44)(43,48);(4,12)(8,13)(11,15)(14,16)(19,28)(22,31)(24,32)(26,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=g,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, 8314046);