#8731151

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

4012 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=g,f^2=g,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, 8638347);