#8534614

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

4264 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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^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, 8441810);