#8424851

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gh,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, 8332047);