#8192597

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	#48173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099793);