#9065945

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	#44148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973141);