#8107223

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	#45398	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014419);