#8814013

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,5,10,8)(4,9,12,14)(11,13,16,15)(17,18)(19,24)(20,26)(21,22)(23,25)(27,32)(28,29)(30,31)(34,37)(39,42)(40,45)(46,48);(2,7)(4,12)(5,8)(11,16)(17,19,23,20)(18,24,25,26)(21,30,27,28)(22,31,32,29)(36,44)(40,45)(41,47)(46,48);(2,7)(3,10)(4,12)(13,15)(18,25)(19,20)(21,27)(29,31)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31)(33,36,38,44)(34,40,37,45)(35,41,43,47)(39,46,42,48);(1,5,6,8)(2,3,7,10)(4,13,12,15)(9,11,14,16)(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8721209);