#8459077

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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)(3,7)(4,5)(6,8)(10,14)(11,17)(16,19)(18,23)(25,26)(27,29)(28,32)(30,33)(31,35)(34,39)(36,37)(38,40);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,30,35)(26,29,33,31)(28,36,38,34)(32,37,40,39);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,36,40,34)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=ghi,f^2=gi,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, 8366273);