#7945610

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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)(9,10,15,16)(11,13,19,23)(12,21,20,17)(14,18,24,22)(25,26)(27,31)(28,38)(29,33)(30,34)(32,36)(35,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11,15,19)(10,13,16,23)(12,14,20,24)(17,22,21,18)(25,27,30,37)(26,31,34,35)(28,29,36,40)(32,39,38,33);(10,16)(11,19)(12,20)(18,22)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,38)(39,40);(9,12,15,20)(10,17,16,21)(11,14,19,24)(13,22,23,18)(25,28,30,36)(26,32,34,38)(27,29,37,40)(31,39,35,33);(1,4,5,2)(3,7,8,6)(9,13,15,23)(10,11,16,19)(12,22,20,18)(14,21,24,17)(25,26)(27,31)(28,32)(29,39)(30,34)(33,40)(35,37)(36,38);(9,14)(10,18)(11,20)(12,19)(13,17)(15,24)(16,22)(21,23)(25,29)(26,33)(27,36)(28,37)(30,40)(31,32)(34,39)(35,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7852806);