#9203669

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	#43971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31)(33,34,40,43)(35,42,39,37)(36,45,38,41)(44,48,47,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,35,40,39)(34,37,43,42)(36,44,38,47)(41,48,45,46);(3,10)(7,8)(11,13)(12,16)(20,28)(24,32)(27,29)(30,31)(33,36,40,38)(34,41,43,45)(35,44,39,47)(37,48,42,46);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,29)(26,32)(33,37,40,42)(34,35,43,39)(36,46,38,48)(41,47,45,44);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,19)(18,21)(20,29)(22,25)(23,26)(24,31)(27,28)(30,32)(33,38,40,36)(34,41,43,45)(35,44,39,47)(37,46,42,48);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,21)(18,25)(19,26)(20,30)(22,23)(24,29)(27,32)(28,31)(33,39,40,35)(34,42,43,37)(36,47,38,44)(41,46,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,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, 9110865);