#8750720

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3747 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)(3,13)(4,10)(5,6)(7,12)(8,15)(9,16)(11,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,8)(4,5)(6,9)(7,15)(10,11)(12,13)(14,16)(18,22)(23,25)(24,29)(30,31)(36,44)(39,46)(42,48)(45,47);(1,4,7,16)(2,9,12,10)(3,5,15,14)(6,13,11,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4)(2,10)(3,14)(5,15)(6,8)(7,16)(9,12)(11,13)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,5,7,14)(2,11,12,6)(3,4,15,16)(8,10,13,9)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,6,7,11)(2,5,12,14)(3,9,15,10)(4,13,16,8)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657916);