#8664418

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

4271 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8571614);