#7987871

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895067);