#8115510

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	#42883	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4510 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:

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

Polycyclic presentation:

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

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, 8022706);