#9040307

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	#48696	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) = 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)(33,34)(35,36)(37,43)(38,40)(39,41)(42,44)(45,46)(47,48);(2,7)(3,9)(8,15)(11,13)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35)(34,37)(36,41)(38,46)(39,43)(40,44)(42,47)(45,48);(2,7)(4,12)(5,10)(11,13)(18,22)(20,29)(23,25)(26,30)(33,36,39,37)(34,35,41,43)(38,45,47,44)(40,46,48,42);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,39,36)(34,35,41,43)(38,44,47,45)(40,46,48,42);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38,39,47)(34,40,41,48)(35,42,43,46)(36,44,37,45);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,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, 8947503);