#8258118

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	#38823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8165314);