#8258092

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	#42424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

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

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, 8165288);