#8158914

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066110);