#8374209

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

4058 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8281405);