#7895571

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	#41349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,36)(28,39)(29,30)(31,35)(32,38)(33,40)(34,37);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,31,38)(26,32,35,36)(28,37,40,29)(30,33,34,39);(10,16)(15,17)(18,23)(22,24)(25,28,31,40)(26,33,35,39)(27,29,38,37)(30,36,34,32);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28)(26,33)(27,37)(29,38)(30,36)(31,40)(32,34)(35,39);(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,29,31,37)(26,34,35,30)(27,28,38,40)(32,39,36,33);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30,31,34)(26,29,35,37)(27,33,38,39)(28,36,40,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=h,f^2=g,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, 7802767);