#7754829

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

4531 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)(9,10,14,16)(11,17,20,15)(12,13,19,18)(21,22,23,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,20)(10,15,16,17)(12,23,19,21)(13,22,18,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(10,16)(11,20)(13,18)(21,23)(26,28)(32,33)(34,36)(39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,24)(19,23)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,12,14,19)(10,18,16,13)(11,21,20,23)(15,22,17,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,18)(10,19,16,12)(11,22,20,24)(15,21,17,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7662025);