#9481512

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

3915 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388708);