#7805061

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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:

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

Polycyclic presentation:

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

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, 7712257);