#8581805

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

4103 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)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,32)(30,31)(33,34,39,36)(35,37,40,42)(38,41,46,44)(43,45,47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(3,9)(6,13)(10,15)(14,16)(18,23)(19,26)(25,31)(27,32)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8489001);