#7654012

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,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, 7561208);