#8750732

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8657928);