#9040306

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	#35437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947502);