#7709341

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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:

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

Polycyclic presentation:

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

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, 7616537);