#7658688

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 7565884);