#10529839

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

4143 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)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(2,7)(3,9)(10,15)(11,16)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(33,34,38,41)(35,42,36,39)(37,40,46,45)(43,48,44,47);(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,36,38,35)(34,39,41,42)(37,43,46,44)(40,48,45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(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=h,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 10437035);