#8975419

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	#54816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4535 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,21,24,18)(25,26,31,29)(27,32,30,37)(28,33,35,38)(34,39,36,40);(2,6)(3,8)(10,15)(12,23)(16,19)(18,21)(25,27,31,30)(26,32,29,37)(28,34,35,36)(33,39,38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,20)(12,23)(13,24)(26,29)(27,30)(28,35)(39,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,28)(26,33)(27,34)(29,38)(30,36)(31,35)(32,39)(37,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,29,31,26)(27,32,30,37)(28,33,35,38)(34,40,36,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,30,31,27)(26,32,29,37)(28,36,35,34)(33,39,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8882615);