#8119658

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(4,8,12,13)(10,15,16,14)(20,28)(24,29)(26,32)(30,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(2,6)(7,9)(8,13)(14,15)(18,22)(23,25)(24,29)(30,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=i,f^2=g,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, 8026854);