#7653239

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 7560435);