#8057936

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

4175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965132);