#8112538

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^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, 8019734);