#8749678

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

4007 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)(4,10)(5,8)(6,12)(7,9)(13,16)(14,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(35,37)(36,40)(43,45)(44,46);(1,3,7,5)(2,8,9,11)(4,15,12,13)(6,14,10,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(2,9)(4,12)(8,11)(13,15)(20,28)(24,31)(27,29)(30,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4)(2,10)(3,13)(5,15)(6,9)(7,12)(8,14)(11,16)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,5,7,3)(2,11,9,8)(4,15,12,13)(6,14,10,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,6,7,10)(2,12,9,4)(3,14,5,16)(8,13,11,15)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=h,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, 8656874);