#7739363

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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)(9,10,14,18)(11,19,22,15)(12,13,17,16)(20,21,23,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,18,19)(12,20,17,23)(13,24,16,21)(26,31)(29,37)(32,33)(35,36);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,23)(13,21)(14,22)(16,24)(17,20)(18,19)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,39)(37,38);(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(9,12,14,17)(10,16,18,13)(11,20,22,23)(15,21,19,24)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(9,13,14,16)(10,17,18,12)(11,21,22,24)(15,20,19,23)(25,29,30,37)(26,34,31,38)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646559);