#8158907

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	#55496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(8,12)(9,14)(10,16)(11,15)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32)(34,38)(36,44)(40,42)(43,45);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,34,39,38)(35,40,37,42)(36,41,44,46)(43,47,45,48);(2,7)(3,8)(9,13)(10,15)(18,24)(20,25)(26,30)(28,32)(33,35,39,37)(34,40,38,42)(36,45,44,43)(41,48,46,47);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37,39,35)(34,42,38,40)(36,43,44,45)(41,47,46,48);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,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, 8066103);