#8359867

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=ghi,f^2=g,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, 8267063);