#10246999

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	#56489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154195);