#8077432

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	#43013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 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)(9,10,14,13)(11,17,18,15)(12,16,21,22)(19,24,20,23)(26,30)(31,33)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,18)(16,24)(19,21)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(11,18)(15,17)(19,20)(23,24)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(9,11)(10,15)(12,20)(13,17)(14,18)(16,24)(19,21)(22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,17,18,15)(12,16,21,22)(19,23,20,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,f^2=g,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, 7984628);