#8630815

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

4031 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,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^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538011);