#8359869

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3780 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8267065);