#7654041

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=h,f^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, 7561237);