#7654204

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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)(25,26,31,29)(27,35,38,32)(28,33,37,39)(30,34,36,40);(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)(26,32)(28,36)(29,35)(30,37)(31,38)(33,34)(39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,26,31,29)(27,35,38,32)(28,33,37,39)(30,34,36,40);(1,3,5,4)(2,7,6,8)(9,13,14,11)(10,17,19,15)(12,21,22,20)(16,24,18,23)(25,28,31,37)(26,33,29,39)(27,36,38,30)(32,34,35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,20)(13,21)(14,22)(15,24)(16,19)(17,23)(25,29,31,26)(27,35,38,32)(28,39,37,33)(30,34,36,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,30,31,36)(26,34,29,40)(27,37,38,28)(32,33,35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=gh,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, 7561400);