#7715449

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	#38914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(11,19)(12,18)(13,15)(14,16)(17,22)(20,23)(21,24)(25,26,31,34)(27,32,30,35)(28,33,39,29)(36,37,38,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,13,19)(12,21,22,20)(17,24,18,23)(25,26)(27,35)(28,29)(30,32)(31,34)(33,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,13,18)(11,20,14,21)(16,23,19,24)(25,27,31,30)(26,32,34,35)(28,36,39,38)(29,40,33,37);(9,12)(10,18)(11,20)(13,17)(14,21)(15,22)(16,24)(19,23)(25,28)(26,33)(27,36)(29,34)(30,38)(31,39)(32,37)(35,40);(1,4,5,3)(2,8,6,7)(9,13,15,10)(11,19,14,16)(12,17,22,18)(20,23,21,24)(25,29,31,33)(26,28,34,39)(27,37,30,40)(32,38,35,36);(9,14,15,11)(10,19,13,16)(12,21,22,20)(17,24,18,23)(25,30,31,27)(26,32,34,35)(28,38,39,36)(29,40,33,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,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, 7622645);