#7952510

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(11,13)(12,20)(14,15)(16,18)(17,22)(19,21)(23,24)(26,30)(28,36)(31,33)(34,40);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,18,13)(12,17,22,20)(19,21,24,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(10,15)(11,18)(17,20)(19,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,12)(10,17)(11,19)(13,23)(14,22)(15,20)(16,21)(18,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,24,20,19)

Polycyclic presentation:

2^9|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, 7859706);