#8110588

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4506 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017784);