#7716105

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	#44976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623301);