#8475829

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	#47951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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)(3,8)(4,6)(5,7)(10,14)(11,17)(16,21)(18,23)(28,36)(32,37)(35,40)(38,39);(1,3,6,5)(2,7,4,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26,30,34)(27,29,33,31)(28,37,36,32)(35,39,40,38);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,2,6,4)(3,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,4,6,2)(3,7,5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,5,6,3)(2,7,4,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,33)(27,34)(28,38)(30,31)(32,40)(35,37)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8383025);