#10096401

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10003597);