#8741936

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

4098 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=gh,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, 8649132);