#8011976

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	#43932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4525 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7919172);