#8037843

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	#40521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4415 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)(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,23,20,21)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,23,24,21)(27,33)(30,37)(34,39)(38,40);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(18,24)(19,20)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,24,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,23,16,21)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(9,14)(10,11)(12,17)(13,21)(15,16)(18,20)(19,24)(22,23)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7945039);