#7654001

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 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)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,2,5,6)(3,7,4,8)(27,34)(31,38)(35,39)(36,40);(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(1,3,5,4)(2,7,6,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=gh,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, 7561197);