#8271302

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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:

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

Polycyclic presentation:

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

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, 8178498);