#8682229

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	#43095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4757 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)(9,10,14,18)(11,13,17,15)(12,16,21,23)(19,22,24,20)(26,30)(27,33)(32,35)(34,36);(3,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,20)(21,24)(22,23)(26,30)(31,38)(32,35)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,18,23)(11,19,17,24)(13,22,15,20)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,23)(21,24)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(9,13)(10,17)(11,18)(12,20)(14,15)(16,19)(21,22)(23,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=g,d^2=g,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, 8589425);