#8908123

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

4446 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,18,13,16)(12,21,22,14)(19,20,23,24)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,39,36);(1,2,6,8)(3,5,7,4)(9,11,15,13)(10,16,17,18)(12,19,22,23)(14,20,21,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,37)(34,38)(39,40);(1,3,6,7)(2,4,8,5)(12,22)(14,21)(19,23)(20,24)(27,35)(28,37)(31,38)(32,40);(1,4,6,5)(2,3,8,7)(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28)(26,32)(27,36)(29,31)(30,37)(33,35)(34,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,17,18)(12,19,22,23)(14,24,21,20)(25,29)(26,33)(27,32)(28,38)(30,39)(31,37)(34,36)(35,40);(1,5,6,4)(2,7,8,3)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,27)(26,31)(28,36)(29,32)(30,35)(33,37)(34,38)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815319);