#8479532

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

4043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386728);