#8080805

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	#48155	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(10,15)(12,17)(16,23)(20,24)(25,26,31,30)(27,32,28,33)(29,34,38,37)(35,39,36,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,17,19,12)(13,16,22,23)(18,24,21,20)(25,27,31,28)(26,32,30,33)(29,35,38,36)(34,39,37,40);(10,15)(12,17)(16,23)(20,24)(25,27,31,28)(26,33,30,32)(29,35,38,36)(34,40,37,39);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,12,15,17)(13,21,22,18)(16,24,23,20)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,15,23)(11,18,19,21)(12,20,17,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988001);