#8424835

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	#39262	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(27,33)(31,35)(34,39)(38,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,16)(10,12,18,13)(14,19,23,24)(17,20,22,21)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(1,3,5,6)(2,4,7,8)(9,12,15,13)(10,11,18,16)(14,20,23,21)(17,19,22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,21)(20,24)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(9,13,15,12)(10,16,18,11)(14,21,23,20)(17,24,22,19)(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)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(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=gh,c^2=gi,d^2=h,e^2=g,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, 8332031);