#7715865

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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)(9,10,15,12)(11,19,20,16)(13,17,21,22)(14,23,24,18)(25,26)(27,32)(28,29)(30,38)(31,34)(33,39)(35,37)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,24)(14,21)(15,20)(17,23)(18,22)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(11,20)(13,21)(16,19)(17,22)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,19,20,16)(13,22,21,17)(14,23,24,18)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(1,4,5,3)(2,7,8,6)(9,13,15,21)(10,17,12,22)(11,14,20,24)(16,18,19,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38);(9,14)(10,18)(11,21)(12,23)(13,20)(15,24)(16,22)(17,19)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^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, 7623061);