#7972476

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	#40509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4582 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)(11,16)(12,21)(13,15)(14,18)(17,22)(19,20)(23,24)(27,33)(30,37)(34,39)(38,40);(1,2,4,5)(3,6,7,8)(9,11,15,19)(10,16,13,20)(12,14,22,24)(17,23,21,18)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(10,13)(11,19)(12,22)(18,23)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,12)(10,17)(11,14)(13,21)(15,22)(16,23)(18,20)(19,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,13,15,10)(11,20,19,16)(12,21,22,17)(14,18,24,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(9,14)(10,18)(11,12)(13,23)(15,24)(16,21)(17,20)(19,22)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,e^2=h,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, 7879672);