#10521338

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

4352 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)(3,8,11,13)(4,9,14,16)(5,10,15,12)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34)(35,42)(36,45)(37,40)(38,44)(39,41)(43,47)(46,48);(2,7)(3,11)(9,16)(10,12)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(33,35)(34,38)(36,46)(37,43)(39,44)(40,48)(41,42)(45,47);(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,36,39,37)(34,40,41,45)(35,43,44,46)(38,47,42,48);(1,3)(2,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,35)(34,38)(36,43)(37,46)(39,44)(40,47)(41,42)(45,48);(1,4)(2,9)(3,12)(5,8)(6,14)(7,16)(10,11)(13,15)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,37,39,36)(34,40,41,45)(35,43,44,46)(38,48,42,47);(1,5)(2,10)(3,9)(4,13)(6,15)(7,12)(8,14)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38)(34,35)(36,47)(37,48)(39,42)(40,43)(41,44)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=hi,e^2=i,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, 10428534);