#8445615

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	#58246	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352811);