#9481488

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=i,e^2=h,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, 9388684);