#7987973

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	#41311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(9,10,15,17)(11,14,13,16)(12,20,21,18)(19,24,22,23)(27,34)(32,38)(35,40)(36,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,14)(12,19,21,22)(18,24,20,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(12,21)(23,24)(26,31)(28,37)(32,38)(35,40);(1,3,5,4)(2,7,6,8)(9,12,15,21)(10,18,17,20)(11,19,13,22)(14,23,16,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(9,13,15,11)(10,14,17,16)(12,22,21,19)(18,23,20,24)(25,28,30,37)(26,33,31,29)(27,35,34,40)(32,39,38,36);(1,4,5,3)(2,8,6,7)(9,14)(10,13)(11,17)(12,23)(15,16)(18,22)(19,20)(21,24)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7895169);