#8271886

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gi,e^2=ghi,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, 8179082);