#9039379

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
prime radical	#54947	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946575);