#10460566

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

4333 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)(3,8,10,5)(4,9,13,15)(11,12,16,14)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,48);(2,7)(3,10)(9,15)(11,16)(19,20)(23,24)(27,28)(31,32)(33,35,39,40)(34,37,42,36)(38,43,47,48)(41,45,46,44);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,3)(2,8)(4,11)(5,7)(6,10)(9,12)(13,16)(14,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,48,46,43);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,44)(45,48);(1,5,6,8)(2,3,7,10)(4,12,13,14)(9,16,15,11)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,38,39,47)(34,41,42,46)(35,43,40,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,f^2=g,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, 10367762);