#10349120

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

4040 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10256316);