#8425222

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	#59018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332418);