#7954814

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

4290 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=hi,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, 7862010);