#8110817

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

4154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8018013);