#8372191

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

4207 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:

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

Polycyclic presentation:

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

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, 8279387);