#7954408

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	#42643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,e^2=h,f^2=i,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, 7861604);