#8118871

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
prime radical	#53545	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(21,29)(24,31)(27,28)(30,32)(34,37)(39,42)(40,45)(47,48);(2,7)(3,9)(10,13)(11,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(2,7)(4,8)(10,13)(12,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35,38,43)(34,39,37,42)(36,44,46,41)(40,48,45,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,36,38,46)(34,40,37,45)(35,41,43,44)(39,47,42,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,34)(35,42,43,39)(36,45,46,40)(41,48,44,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026067);