#8075210

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	#47830	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4708 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,f^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, 7982406);