#10412122

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319318);