#7715500

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622696);