#8739321

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

4092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(35,37)(36,40)(43,45)(44,46);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(17,19,23,21)(18,24,26,27)(20,28,25,30)(22,29,32,31)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(2,9)(4,10)(8,11)(12,14)(17,20,23,25)(18,22,26,32)(19,28,21,30)(24,29,27,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,32)(27,31)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,21,23,19)(18,24,26,27)(20,30,25,28)(22,29,32,31)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(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=gh,d^2=h,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, 8646517);