#7779412

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	#44975	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 7686608);