#8471843

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

4239 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8379039);