#7901812

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	#35682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 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:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=h,f^2=i,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, 7809008);