#8118627

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	#46405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8025823);