#7792180

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	#37839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4523 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7699376);