#7735615

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	#37049	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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:

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

Polycyclic presentation:

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

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, 7642811);