#7645389

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	#41368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(11,19)(12,18)(13,15)(14,16)(17,22)(20,23)(21,24)(29,37)(32,39)(35,38)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,19)(12,20,22,21)(17,23,18,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,12)(10,17)(11,20)(13,18)(14,21)(15,22)(16,23)(19,24)(25,27,30,31)(26,28,33,34)(29,35,37,38)(32,36,39,40);(9,13,15,10)(11,19,14,16)(12,18,22,17)(20,24,21,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,12)(10,18)(11,20)(13,17)(14,21)(15,22)(16,24)(19,23)(25,28)(26,31)(27,33)(29,36)(30,34)(32,38)(35,39)(37,40);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,19,13,16)(12,21,22,20)(17,24,18,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552585);