#7659052

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566248);