#8093703

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000899);