#8183761

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	#52313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4621 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26,30,34)(27,35,37,31)(28,38,36,32)(29,33,39,40);(2,6)(3,8)(10,11)(15,17)(16,18)(22,23)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,33,38,40);(1,3)(2,4)(5,8)(6,7)(13,21)(16,18)(19,24)(22,23)(25,28)(26,32)(27,29)(30,36)(31,33)(34,38)(35,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,33)(27,36)(28,37)(30,39)(31,38)(32,35)(34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090957);