#8206123

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

4238 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)(3,11,12,8)(4,5,10,9)(13,15,16,14)(33,34,39,42)(35,40,46,41)(36,47,37,38)(43,45,44,48);(2,7)(3,12)(4,10)(14,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,35,39,46)(34,40,42,41)(36,44,37,43)(38,45,47,48);(2,7)(5,9)(8,11)(14,15)(18,22)(23,29)(24,27)(30,31)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,48,41,45);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,48,41,45);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,15,11,14)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)(33,35)(34,41)(36,43)(37,44)(38,45)(39,46)(40,42)(47,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,16,11,13)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,38)(34,36)(35,45)(37,42)(39,47)(40,44)(41,43)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,e^2=g,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

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, 8113319);