#10412105

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

4225 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10319301);