#9040462

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	#50762	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(3,10,9,5)(4,8,12,15)(11,16,13,14)(17,18,21,23)(19,22,25,30)(20,24,28,31)(26,29,27,32);(2,7)(3,9)(8,15)(11,13)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,7)(4,12)(5,10)(11,13)(18,23)(19,25)(24,31)(26,27)(34,38)(36,45)(39,41)(42,46);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,20,21,28)(18,24,23,31)(19,26,25,27)(22,29,30,32)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,32)(28,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=h,d^2=h,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, 8947658);