#7898506

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	#44776	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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)(11,20)(12,14)(13,23)(15,17)(16,21)(18,24)(19,22)(26,32)(28,34)(33,35)(36,38);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,17,20)(12,22,24,13)(14,23,18,19)(25,26,31,32)(27,35,29,33)(28,30,34,40)(36,39,38,37);(10,17)(11,21)(12,24)(19,23)(25,27,31,29)(26,33,32,35)(28,36,34,38)(30,37,40,39);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,22)(13,21)(14,17)(15,24)(16,19)(20,23)(25,28,31,34)(26,30,32,40)(27,36,29,38)(33,37,35,39);(1,2,5,4)(3,6,8,7)(9,13,15,22)(10,19,17,23)(11,12,21,24)(14,20,18,16)(25,29,31,27)(26,33,32,35)(28,38,34,36)(30,37,40,39);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,24)(11,23,21,19)(13,16,22,20)(25,30)(26,34)(27,37)(28,32)(29,39)(31,40)(33,38)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805702);