#8075604

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	#44931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4483 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)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,26,30,31)(27,33,34,28)(29,37,38,32)(35,36,40,39);(1,2,5,6)(3,7,4,8)(9,11,14,20)(10,15,16,22)(12,13,19,21)(17,18,23,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,16)(15,22)(17,23)(18,24)(26,31)(27,34)(29,38)(36,39);(9,12)(10,17)(11,13)(14,19)(15,18)(16,23)(20,21)(22,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,23)(16,24)(17,22)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27)(26,33)(28,31)(29,35)(30,34)(32,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,e^2=g,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, 7982800);