#7736375

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7643571);