#8455658

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	#46035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8362854);