#7719918

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	#34609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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)(10,13)(11,17)(16,21)(19,20)(28,36)(32,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,13)(11,18,17,15)(12,16,22,21)(19,24,20,23)(26,30)(27,33)(32,37)(35,39);(1,3,4,7)(2,6,5,8)(9,11,14,17)(10,15,13,18)(12,20,22,19)(16,24,21,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,12,14,22)(10,16,13,21)(11,19,17,20)(15,23,18,24);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,13,14,10)(11,18,17,15)(12,21,22,16)(19,24,20,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^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, 7627114);