#8687334

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	#53499	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8594530);