#8657958

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	#57964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4547 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565154);