#9166765

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	#48442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34,38,40)(35,36,42,44)(37,39,48,47)(41,43,45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35)(34,36)(37,45)(38,42)(39,46)(40,44)(41,48)(43,47);(3,10)(6,7)(11,16)(14,15)(18,20)(23,25)(24,28)(31,32)(33,36,38,44)(34,35,40,42)(37,43,48,46)(39,41,47,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36)(34,35)(37,46)(38,44)(39,45)(40,42)(41,47)(43,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,39)(35,41)(36,43)(38,48)(40,47)(42,45)(44,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,34)(35,36)(37,47)(38,40)(39,48)(41,46)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^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, 9073961);