#7746082

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	#44836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653278);