#8757087

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664283);