#9203623

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	#42057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gh,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, 9110819);