#8115490

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

3835 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022686);