#8436921

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	#58242	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8344117);