#8406653

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

4174 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8313849);