#8011985

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	#44226	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=gh,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, 7919181);