#8107532

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

4575 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014728);