#8073110

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	#43039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,26)(27,34)(28,37)(29,31)(30,33)(32,38)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,15)(10,12,17,19)(13,18,22,23)(16,20,21,24)(25,27,30,29)(26,31,33,34)(28,36,38,35)(32,40,37,39);(11,15)(12,19)(18,23)(20,24)(25,28,30,38)(26,32,33,37)(27,35,29,36)(31,39,34,40);(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,24)(20,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,40)(37,38);(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24)(25,28)(26,32)(27,36)(29,35)(30,38)(31,40)(33,37)(34,39);(1,3,4,7)(2,5,6,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,30,27)(26,31,33,34)(28,35,38,36)(32,40,37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,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, 7980306);