#8159573

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,f^2=g,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, 8066769);