#8528888

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

4048 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)(3,8)(4,11)(5,9)(6,7)(10,16)(12,13)(14,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(1,2)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,7)(4,8)(9,15)(12,16)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,12)(10,14)(15,16)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,37)(34,40)(35,43)(36,41)(38,45)(39,46)(42,47)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,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, 8436084);