#10451615

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	#39525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358811);