#7720498

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	#37379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627694);