#9170710

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	#52969	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^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, 9077906);