#8136884

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	#50958	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4465 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:

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

Polycyclic presentation:

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

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, 8044080);