#8479516

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

4170 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)(3,9,4,13)(5,14,15,10)(6,11,16,12)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,38,42)(35,41,45,39)(36,46,37,40)(43,47,44,48);(2,8)(5,15)(9,13)(11,12)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,38,45)(34,39,42,41)(36,43,37,44)(40,47,46,48);(1,3,7,4)(2,9,8,13)(5,12,15,11)(6,10,16,14)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,37)(34,40,42,46)(35,43,45,44)(39,47,41,48);(1,4,7,3)(2,9,8,13)(5,11,15,12)(6,10,16,14)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,34)(35,41)(36,40)(37,46)(38,42)(39,45)(43,48)(44,47);(1,5)(2,10)(3,12)(4,11)(6,13)(7,15)(8,14)(9,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,48)(46,47);(1,6,7,16)(2,11,8,12)(3,10,4,14)(5,13,15,9)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,38,36)(34,40,42,46)(35,44,45,43)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,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, 8386712);