#7715533

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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)(10,13)(11,14)(17,20)(19,21)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,36)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10,15,13)(11,18,14,16)(12,20,22,17)(19,23,21,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3,5,4)(2,6,8,7)(9,11,15,14)(10,16,13,18)(12,21,22,19)(17,24,20,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,4,5,3)(2,7,8,6)(9,12,15,22)(10,17,13,20)(11,19,14,21)(16,23,18,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(9,13,15,10)(11,16,14,18)(12,20,22,17)(19,23,21,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14,15,11)(10,16,13,18)(12,21,22,19)(17,23,20,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622729);