#8425019

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

4578 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332215);