#8159618

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	#41090	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=h,f^2=g,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, 8066814);