#7805066

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26,29,30)(27,31,33,34)(28,36,37,32)(35,40,39,38);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,14,17,16)(12,20,21,19)(15,23,24,22)(26,30)(27,33)(28,37)(38,40);(11,18)(14,16)(19,20)(22,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,12,13,21)(10,15,17,24)(11,19,18,20)(14,22,16,23)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40);(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,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, 7712262);