#8193879

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	#51768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,14)(15,16)(33,34)(35,36)(37,45)(38,42)(39,40)(41,47)(43,48)(44,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,36,40,42)(37,43,47,46)(41,44,45,48);(2,7)(6,9)(8,12)(14,15)(18,22)(23,25)(24,29)(30,31)(34,40)(35,38)(41,45)(43,46);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,39,42)(34,35,40,38)(37,44,47,48)(41,43,45,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,39,47)(34,41,40,45)(35,43,38,46)(36,44,42,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,38,39,35)(34,42,40,36)(37,46,47,43)(41,48,45,44)

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=i,f^2=gh,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, 8101075);