#8682236

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	#40588	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4668 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)(10,14)(11,17)(16,19)(18,20)(25,26,31,36)(27,30,35,32)(28,33,38,40)(29,34,37,39);(3,7)(6,8)(10,14)(15,22)(16,19)(23,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,31,38)(26,33,36,40)(27,37,35,29)(30,39,32,34);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27)(26,30)(28,37)(29,38)(31,35)(32,36)(33,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,29,31,37)(26,34,36,39)(27,38,35,28)(30,40,32,33);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,30)(26,35)(27,36)(28,39)(29,33)(31,32)(34,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^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, 8589432);