#7715860

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	#39166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,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, 7623056);