#8677134

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	#45604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8584330);