#7779399

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	#34880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686595);