#8761966

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	#49763	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669162);