#7650924

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	#41029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4445 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)(11,16)(12,14)(13,19)(15,17)(18,20)(21,22)(23,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,21)(10,16,17,22)(12,13,20,23)(14,19,18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(10,17)(14,18)(16,22)(19,24)(27,33)(31,35)(34,39)(38,40);(9,12)(10,18)(11,13)(14,17)(15,20)(16,24)(19,22)(21,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3,4,7)(2,5,6,8)(9,13)(10,19)(11,20)(12,21)(14,22)(15,23)(16,18)(17,24)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40);(9,14,15,18)(10,12,17,20)(11,19,21,24)(13,22,23,16)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558120);