#8001005

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 7908201);