#8479528

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

4105 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,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, 8386724);