#7766340

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	#36318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4538 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)(9,10,15,12)(11,19,20,16)(13,17,23,21)(14,18,24,22)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,19)(13,14)(15,20)(17,22)(18,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40);(11,20)(14,24)(16,19)(18,22)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(9,13)(10,17)(11,14)(12,21)(15,23)(16,22)(18,19)(20,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(9,14)(10,18)(11,13)(12,22)(15,24)(16,21)(17,19)(20,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=gh,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, 7673536);