#8093823

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	#46274	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001019);