#9243127

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(3,9)(7,13)(10,15)(14,16)(18,20)(21,29)(23,25)(27,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,6)(7,13)(8,11)(14,16)(18,20)(19,26)(24,28)(27,32)(34,35)(39,41)(40,42)(46,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150323);