#7654011

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	#40529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4488 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)(10,16)(11,12)(18,24)(19,20)(25,26,31,35)(27,36,39,32)(28,33,40,29)(30,34,38,37);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,17)(14,18)(15,16)(19,21)(20,22)(23,24)(25,27)(26,32)(28,34)(29,38)(30,33)(31,39)(35,36)(37,40);(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,24,22)(25,28)(26,33)(27,37)(29,35)(30,36)(31,40)(32,38)(34,39);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,22,24,21)(25,29,31,33)(26,28,35,40)(27,38,39,30)(32,34,36,37);(9,13,15,17)(10,11,16,12)(14,22,23,21)(18,20,24,19)(25,28)(26,33)(27,34)(29,35)(30,32)(31,40)(36,38)(37,39);(1,4,5,3)(2,8,6,7)(9,14,15,23)(10,18,16,24)(11,19,12,20)(13,21,17,22)(25,30,31,38)(26,34,35,37)(27,33,39,29)(28,36,40,32)

Polycyclic presentation:

2^9|a^2=i,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, 7561207);