#7915890

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
prime radical	#45450	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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)(9,10,16,22)(11,15,21,17)(12,20,14,18)(13,24,23,19)(26,32)(27,29)(33,39)(35,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,17)(12,13)(14,23)(15,22)(16,21)(18,24)(19,20)(25,26)(27,30)(28,33)(29,34)(31,32)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,16,14)(10,18,22,20)(11,23,21,13)(15,24,17,19)(25,27,31,29)(26,30,32,34)(28,36,38,35)(33,40,39,37);(9,13,16,23)(10,19,22,24)(11,12,21,14)(15,18,17,20)(25,28,31,38)(26,33,32,39)(27,35,29,36)(30,37,34,40);(9,14,16,12)(10,20,22,18)(11,23,21,13)(15,24,17,19)(25,29,31,27)(26,34,32,30)(28,36,38,35)(33,40,39,37);(1,4,5,3)(2,8,6,7)(9,15)(10,21)(11,22)(12,24)(13,18)(14,19)(16,17)(20,23)(25,30,31,34)(26,27,32,29)(28,37,38,40)(33,35,39,36)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7823086);