#8375554

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8282750);