#7657518

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,5,6,8)(9,13,15,11)(10,16,17,18)(12,22,23,19)(14,20,21,24);(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564714);