#8695324

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	#47950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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)(3,6)(4,7)(5,8)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23);(1,3)(2,4)(5,7)(6,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(26,31)(27,33)(32,39)(34,40);(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(1,2,5,8)(3,6,7,4)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,4,5,6)(2,7,8,3)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602520);