#7720501

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	#35129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4569 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)(12,17)(13,15)(14,24)(18,20)(19,23)(21,22)(25,26,30,33)(27,28,31,35)(29,37,38,32)(34,40,39,36);(1,2,4,5)(3,6,7,8)(9,11,15,19)(10,16,13,23)(12,20,21,14)(17,18,22,24)(27,31)(28,35)(34,39)(36,40);(1,3,4,7)(2,6,5,8)(10,13)(11,19)(12,21)(18,24)(25,27)(26,28)(29,34)(30,31)(32,36)(33,35)(37,40)(38,39);(9,12)(10,17)(11,20)(13,22)(14,19)(15,21)(16,18)(23,24);(9,13,15,10)(11,16,19,23)(12,22,21,17)(14,24,20,18)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(9,14,15,20)(10,18,13,24)(11,12,19,21)(16,22,23,17)(25,29,30,38)(26,32,33,37)(27,34,31,39)(28,36,35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^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, 7627697);