#10217032

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10124228);