#8075442

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	#38013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=h,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, 7982638);