#8188051

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	#44976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095247);