#8114369

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

3979 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8021565);