#8202023

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=g,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

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, 8109219);