#8037040

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	#42432	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=g,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, 7944236);