#8829216

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	#40959	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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)(3,9)(4,11)(5,13)(6,7)(8,15)(10,12)(14,16)(19,25)(22,26)(27,31)(30,32)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,45,47,41);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(35,44)(37,47)(39,43)(41,45);(2,7)(4,9)(8,13)(12,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35)(34,39)(36,45)(37,40)(38,44)(41,46)(42,43)(47,48);(1,4,6,9)(2,3,7,11)(5,12,15,16)(8,10,13,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,36)(34,40)(35,41)(37,43)(38,46)(39,47)(42,48)(44,45);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,35)(34,39)(36,45)(37,40)(38,44)(41,46)(42,43)(47,48);(1,5)(2,8)(3,10)(4,12)(6,15)(7,13)(9,16)(11,14)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37)(34,41)(35,40)(36,43)(38,47)(39,46)(42,45)(44,48)

Polycyclic presentation:

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

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, 8736412);