#7764874

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	#45480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,31,33)(27,32,35,37)(28,34,36,40)(29,38,39,30);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27,31,35)(26,32,33,37)(28,29,36,39)(30,34,38,40);(10,17)(12,22)(16,19)(20,23)(26,33)(27,35)(28,36)(30,38);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28)(26,34)(27,29)(30,37)(31,36)(32,38)(33,40)(35,39);(1,3,5,4)(2,6,8,7)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29)(26,30)(27,36)(28,35)(31,39)(32,34)(33,38)(37,40);(1,4,5,3)(2,7,8,6)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30)(26,29)(27,34)(28,37)(31,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7672070);