#9059101

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	#46704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8966297);