#8075408

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	#44013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4282 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)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,40)(37,38);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,16,18)(12,19,20,21)(17,22,23,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,16)(15,18)(17,23)(22,24)(26,29)(27,32)(30,38)(36,39);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,28,31,34)(26,32,29,27)(30,39,38,36)(33,35,37,40);(9,13,14,11)(10,18,16,15)(12,19,20,21)(17,22,23,24)(25,29,31,26)(27,34,32,28)(30,37,38,33)(35,39,40,36);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,40)(34,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982604);