#7654195

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4373 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:

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

Polycyclic presentation:

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

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, 7561391);