#8192593

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4527 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)(3,4)(5,7)(6,8)(9,10,15,18)(11,12,13,16)(14,17,22,24)(19,20,21,23)(25,26,30,32)(27,31,34,28)(29,37,38,33)(35,36,40,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40);(2,7)(4,6)(14,22)(17,24)(19,21)(20,23)(26,32)(27,34)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,28,30,31)(26,27,32,34)(29,36,38,39)(33,35,37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,18,12)(14,21,22,19)(17,23,24,20)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40);(1,4,5,6)(2,3,7,8)(9,14,15,22)(10,17,18,24)(11,19,13,21)(12,20,16,23)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=ghi,f^2=g,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, 8099789);