#7653422

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	#39900	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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)(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,23,24,21)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=i,f^2=i,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, 7560618);