#8875566

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

3962 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gh,e^2=g,f^2=i,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, 8782762);