#8096759

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003955);