#8075570

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	#43362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4412 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)(11,16)(12,17)(13,15)(14,18)(19,21)(20,24)(22,23)(25,26,31,30)(27,29,34,32)(28,33,37,39)(35,38,40,36);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,24)(18,23)(25,27,31,34)(26,32,30,29)(28,35,37,40)(33,36,39,38);(1,2,5,4)(3,6,8,7)(10,13)(11,19)(14,20)(17,23)(27,34)(29,32)(35,40)(36,38);(9,12)(10,17)(11,20)(13,23)(14,19)(15,22)(16,24)(18,21)(25,28)(26,33)(27,35)(29,38)(30,39)(31,37)(32,36)(34,40);(9,13,15,10)(11,21,19,16)(12,23,22,17)(14,24,20,18)(25,29)(26,27)(28,36)(30,34)(31,32)(33,40)(35,39)(37,38);(1,4,5,2)(3,7,8,6)(9,14,15,20)(10,18,13,24)(11,22,19,12)(16,23,21,17)(25,30,31,26)(27,29,34,32)(28,33,37,39)(35,36,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,e^2=h,f^2=gh,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, 7982766);