#7658746

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	#41983	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565942);