#10482456

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10389652);