#7742561

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=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, 7649757);