#8158877

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	#56389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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)(18,22)(19,25)(24,30)(26,27)(33,34)(35,41)(36,39)(37,40)(38,44)(42,46)(43,47)(45,48);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)(34,36)(40,41)(42,44)(47,48);(2,7)(3,8)(9,13)(10,15)(19,25)(23,29)(26,27)(31,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,19,21,25)(18,23,22,29)(20,27,28,26)(24,32,30,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=g,f^2=h,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, 8066073);