#8291618

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198814);