#7709740

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	#45769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,30,31)(27,29,33,38)(28,35,36,32)(34,40,39,37);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,36)(39,40);(10,17)(12,18)(16,19)(20,22)(26,31)(27,33)(28,36)(37,40);(1,3,5,4)(2,6,8,7)(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,28,30,36)(26,32,31,35)(27,34,33,39)(29,37,38,40);(1,4,5,3)(2,7,8,6)(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,29)(26,27)(28,37)(30,38)(31,33)(32,34)(35,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616936);