#7969453

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4571 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)(9,10)(11,15)(12,22)(13,18)(14,16)(17,19)(20,21)(23,24)(26,28)(31,33)(32,36)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,24)(21,22)(25,26,30,28)(27,33,35,31)(29,32,38,36)(34,40,37,39);(10,16)(11,19)(12,18)(21,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,4,5,3)(2,8,6,7)(9,12)(10,13)(11,20)(14,18)(15,23)(16,22)(17,21)(19,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40);(9,13)(10,18)(11,21)(12,16)(14,22)(15,20)(17,24)(19,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876649);