#7659965

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	#41257	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(10,14)(15,21)(16,22)(23,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,4,5,2)(3,7,8,6)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

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, 7567161);