#10320386

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3872 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10227582);