#8528879

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8436075);