#8180496

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	#46601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087692);