#8739325

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

3937 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)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(35,37)(36,40)(43,45)(44,46);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(20,28)(25,30)(27,29)(31,32)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,2)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,3)(2,6)(4,13)(5,11)(7,16)(8,9)(10,14)(12,15)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 8646521);