#8534612

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

4101 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)(3,8)(4,10)(5,14)(6,11)(7,9)(12,16)(13,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,3,7,5)(2,8,9,14)(4,12,13,11)(6,10,16,15)(18,22)(23,25)(24,29)(30,31)(34,39)(40,42)(41,44)(47,48);(2,9)(3,5)(10,15)(11,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,37)(34,40,39,42)(36,45,46,43)(41,48,44,47);(1,4,7,13)(2,10,9,15)(3,11,5,12)(6,14,16,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,3)(2,8,9,14)(4,12,13,11)(6,15,16,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,6)(2,11)(3,10)(4,8)(5,15)(7,16)(9,12)(13,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,35)(34,40,39,42)(36,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,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, 8441808);