#8096172

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	#45443	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4615 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003368);