#8479568

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	#53206	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,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, 8386764);