#7770302

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4569 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)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,24)(25,26,31,33)(27,35,29,32)(28,34,38,30)(36,37,39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,13)(14,19,22,20)(18,23,24,21)(25,27,31,29)(26,32,33,35)(28,36,38,39)(30,37,34,40);(1,3,5,4)(2,7,6,8)(10,17)(11,12)(14,22)(21,23)(26,33)(28,38)(32,35)(36,39);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,16,17,13)(14,20,22,19)(18,23,24,21)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,40)(35,37);(1,4,5,3)(2,8,6,7)(9,13,15,16)(10,12,17,11)(14,21,22,23)(18,20,24,19)(25,29,31,27)(26,35,33,32)(28,39,38,36)(30,40,34,37);(9,14,15,22)(10,18,17,24)(11,19,12,20)(13,21,16,23)(25,30,31,34)(26,28,33,38)(27,37,29,40)(32,36,35,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

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, 7677498);