#7779411

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,32,29)(27,33,31,36)(28,35,30,34)(37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,22)(10,16,20,21)(12,23,18,14)(13,24,17,19)(25,27,32,31)(26,33,29,36)(28,38,30,37)(34,40,35,39);(11,22)(14,23)(16,21)(19,24)(25,28,32,30)(26,34,29,35)(27,37,31,38)(33,39,36,40);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,29,32,26)(27,36,31,33)(28,34,30,35)(37,39,38,40);(1,3,4,7)(2,6,5,8)(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,30,32,28)(26,35,29,34)(27,37,31,38)(33,39,36,40);(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,31,32,27)(26,36,29,33)(28,37,30,38)(34,39,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=i,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, 7686607);