#7732437

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	#37271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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)(9,10,14,15)(11,16,18,13)(12,20,21,17)(19,22,24,23)(25,26)(27,36)(28,33)(29,32)(30,34)(31,35)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,19)(14,18)(15,16)(17,22)(20,23)(21,24)(25,27,31,29)(26,32,35,36)(28,37,34,39)(30,40,33,38);(10,15)(11,18)(12,21)(22,23)(27,29)(32,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,28,31,34)(26,33,35,30)(27,37,29,39)(32,38,36,40);(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,29,31,27)(26,32,35,36)(28,39,34,37)(30,40,33,38);(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,19,20,24)(25,30,31,33)(26,34,35,28)(27,38,29,40)(32,37,36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7639633);