#7853417

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	#47679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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)(9,10,15,19)(11,20,23,16)(12,13,24,17)(14,18,22,21)(27,29)(32,34)(35,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,23)(10,16,19,20)(12,21,24,18)(13,22,17,14)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(9,12,15,24)(10,17,19,13)(11,21,23,18)(14,20,22,16)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,21)(14,20)(15,24)(16,22)(17,19)(18,23)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,22)(14,23)(15,17)(16,21)(18,20)(19,24)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,19,21)(11,13,23,17)(12,20,24,16)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7760613);