#10480969

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

4329 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,40,38)(35,37,41,36)(39,46,47,42)(43,44,48,45);(2,7)(3,9)(10,15)(11,16)(19,20)(23,24)(27,28)(31,32)(33,35)(34,36)(37,38)(39,43)(40,41)(42,44)(45,46)(47,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,40,37)(34,41,38,35)(39,44,47,45)(42,48,46,43);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,37,40,36)(34,41,38,35)(39,45,47,44)(42,48,46,43);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38,40,34)(35,37,41,36)(39,42,47,46)(43,44,48,45);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,39,40,47)(34,42,38,46)(35,43,41,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,e^2=g,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, 10388165);