#8388964

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	#50087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4523 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)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(25,26)(27,33)(28,35)(29,31)(30,32)(34,39)(36,40)(37,38);(2,6)(4,7)(10,13)(15,17)(16,20)(23,24)(29,37)(31,38)(34,40)(36,39);(2,6)(3,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,22,16)(18,24,21,23)(25,29)(26,31)(27,34)(28,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296160);