#8384273

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	#43009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8291469);