#7709742

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	#45770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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)(9,10,15,17)(11,18,13,16)(12,14,21,24)(19,23,22,20)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,18)(12,19,21,22)(14,20,24,23)(25,27,31,30)(26,32,33,35)(28,36,34,38)(29,37,40,39);(10,17)(11,13)(12,21)(20,23)(26,33)(28,34)(32,35)(36,38);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,19)(13,22)(15,21)(16,20)(17,24)(18,23)(25,28,31,34)(26,29,33,40)(27,36,30,38)(32,37,35,39);(1,2,5,4)(3,6,8,7)(9,13,15,11)(10,18,17,16)(12,22,21,19)(14,23,24,20)(25,29)(26,34)(27,37)(28,33)(30,39)(31,40)(32,38)(35,36);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,20)(13,23)(15,24)(16,19)(17,21)(18,22)(25,30,31,27)(26,35,33,32)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7616938);