#8197299

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	#50642	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,3)(2,4)(5,6)(7,8)(9,10,14,13)(11,17,19,15)(12,16,21,20)(18,23,24,22)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(1,4,5,8)(2,6,7,3)(10,13)(15,17)(16,20)(22,23)(26,32)(27,33)(34,37)(35,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(1,3)(2,4)(5,6)(7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,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, 8104495);