#9481482

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	#53350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,30)(31,32)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(3,9)(7,13)(10,15)(14,16)(18,24)(21,28)(25,26)(29,31)(34,35)(40,43)(41,42)(47,48);(2,6)(7,13)(8,11)(14,16)(17,19,23,20)(18,25,24,26)(21,29,28,31)(22,32,27,30)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,26,24,25)(21,29,28,31)(22,32,27,30)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,28)(18,27,24,22)(19,29,20,31)(25,30,26,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,27)(18,28,24,21)(19,30,20,32)(25,29,26,31)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,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, 9388678);