#8749708

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,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, 8656904);