#7941730

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	#38317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,e^2=gh,f^2=h,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, 7848926);