#10247189

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

4002 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154385);