#10320352

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3857 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=hi,e^2=h,f^2=g,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, 10227548);