#9815223

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9722419);