#8749798

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

3772 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)(3,11)(4,5)(6,9)(7,8)(10,15)(12,13)(14,16)(18,22)(19,25)(20,28)(30,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,8)(4,10)(9,11)(12,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,7,6)(2,9,8,11)(4,14,10,12)(5,13,15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,10)(2,5,8,15)(3,12,6,14)(9,16,11,13)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,10)(3,13)(4,8)(6,16)(7,15)(9,12)(11,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48);(1,6,7,3)(2,9,8,11)(4,12,10,14)(5,13,15,16)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,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, 8656994);