#8096117

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	#52117	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4363 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=h,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, 8003313);