#9166732

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(19,26)(21,23)(27,29)(30,32)(33,34,40,44)(35,43,39,37)(36,45,47,41)(38,42,48,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,21)(20,28)(22,25)(23,26)(24,31)(27,30)(29,32)(33,35,40,39)(34,37,44,43)(36,42,47,46)(38,41,48,45);(3,10)(6,7)(11,16)(14,15)(17,19,22,26)(18,23,25,21)(20,27,31,29)(24,30,28,32)(33,36,40,47)(34,41,44,45)(35,42,39,46)(37,48,43,38);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,37,40,43)(34,35,44,39)(36,38,47,48)(41,46,45,42);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,30)(25,28)(26,29)(33,38)(34,42)(35,45)(36,37)(39,41)(40,48)(43,47)(44,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,19)(20,30)(22,23)(24,29)(25,26)(27,28)(31,32)(33,39,40,35)(34,43,44,37)(36,46,47,42)(38,45,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9073928);