#8479533

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

4059 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18)(19,25)(20,24)(21,30)(22,27)(23,28)(26,31)(29,32)(36,44)(39,45)(43,47)(46,48);(2,7)(3,10)(9,12)(11,16)(17,19,23,20)(18,24,28,25)(21,29,31,27)(22,26,32,30)(33,34)(35,42)(36,45)(37,40)(38,41)(39,44)(43,46)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,23,20)(18,25,28,24)(21,27,31,29)(22,26,32,30)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,46,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,28,24)(21,29,31,27)(22,26,32,30)(33,34)(35,37)(36,39)(38,41)(40,42)(43,46)(44,45)(47,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21,23,31)(18,26,28,30)(19,27,20,29)(22,25,32,24)(33,36)(34,39)(35,43)(37,46)(38,44)(40,47)(41,45)(42,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,32)(18,27,28,29)(19,26,20,30)(21,25,31,24)(33,37)(34,40)(35,41)(36,46)(38,42)(39,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=gh,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, 8386729);