#9214729

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121925);