#7642003

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4404 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)(3,7,8,5)(13,21)(18,22)(19,24)(20,23)(26,31)(27,29)(33,38)(35,37);(1,3,4,8)(2,5,6,7)(10,15)(12,16)(18,22)(20,23)(28,36)(33,38)(35,37)(39,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,30,27)(26,34,31,32)(28,37,36,35)(33,40,38,39)

Polycyclic presentation:

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

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, 7549199);