#8184777

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	#45663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,24,21)(19,25,22,31)(20,29,28,26)(23,27,30,32)(33,34)(35,45)(36,48)(37,43)(38,42)(39,44)(40,47)(41,46);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,19,24,22)(18,25,21,31)(20,23,28,30)(26,32,29,27)(33,35)(34,40)(36,37)(38,46)(39,47)(41,43)(42,48)(44,45);(2,7)(5,8)(9,12)(13,15)(17,20)(18,26)(19,23)(21,29)(22,30)(24,28)(25,32)(27,31)(33,36,39,46)(34,41,44,48)(35,37,47,38)(40,43,45,42);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,21,24,18)(19,25,22,31)(20,29,28,26)(23,32,30,27)(33,37,39,38)(34,42,44,43)(35,36,47,46)(40,48,45,41);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,22,24,19)(18,25,21,31)(20,30,28,23)(26,32,29,27)(33,38,39,37)(34,43,44,42)(35,46,47,36)(40,41,45,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,23,24,30)(18,27,21,32)(19,28,22,20)(25,26,31,29)(33,34)(35,40)(36,48)(37,42)(38,43)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,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, 8091973);