#7720343

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	#37283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,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, 7627539);