#7720052

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 7627248);