#8739348

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

4084 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8646544);