#8158945

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=g,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,d^c=dgi,e^d=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, 8066141);