#8581803

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gi,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, 8488999);