#10451612

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	#56310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,24)(20,23)(21,22)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(9,11,15,12)(10,17,13,16)(14,19,22,20)(18,24,21,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,12,15,11)(10,17,13,16)(14,19,22,20)(18,23,21,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,13,15,10)(11,16,12,17)(14,21,22,18)(19,23,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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, 10358808);