#9481412

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	#53340	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=h,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, 9388608);