#7714797

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	#38590	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=gi,f^2=g,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, 7621993);