#7988165

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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)(9,10,13,14)(11,17,18,15)(12,16,21,20)(19,23,24,22)(27,34)(32,38)(35,40)(36,39);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(10,14)(11,18)(16,20)(19,24)(26,31)(28,37)(32,38)(35,40);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,28,30,37)(26,33,31,29)(27,35,34,40)(32,39,38,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(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=gh,c^2=gh,d^2=g,e^2=hi,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, 7895361);