#8751293

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

3763 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(35,41)(36,43)(38,44)(39,46);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(18,19)(23,25)(24,26)(30,31)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,9)(4,14)(8,12)(11,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=i,e^2=gh,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, 8658489);