#7736673

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	#44157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 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)(9,10,14,19)(11,16,20,15)(12,17,22,23)(13,18,24,21)(25,26)(27,28)(29,33)(30,32)(31,34)(35,36)(37,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,24)(18,22)(25,27)(26,29)(28,30)(31,39)(32,33)(34,36)(35,40)(37,38);(11,20)(12,22)(15,16)(17,23)(25,28,32,29)(26,33,30,27)(31,36,40,37)(34,39,38,35);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,29,32,28)(26,33,30,27)(31,37,40,36)(34,39,38,35);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,17,19,23)(11,21,20,18)(13,15,24,16)(25,30,32,26)(27,28,33,29)(31,38,40,34)(35,36,39,37);(9,13)(10,18)(11,17)(12,15)(14,24)(16,22)(19,21)(20,23)(25,31,32,40)(26,34,30,38)(27,35,33,39)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643869);