#8197218

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	#53658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104414);