#8001006

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

4461 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7908202);