#7734383

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	#43021	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4561 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)(9,10,15,14)(11,18,13,16)(12,17,22,21)(19,24,20,23)(25,26,31,35)(27,36,39,32)(28,33,40,30)(29,37,38,34);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,14,18)(12,20,22,19)(17,24,21,23)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,28,31,40)(26,30,35,33)(27,37,39,34)(29,32,38,36);(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,24)(18,23)(25,28)(26,33)(27,37)(29,32)(30,35)(31,40)(34,39)(36,38);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,18)(12,19,22,20)(17,24,21,23)(25,29,31,38)(26,34,35,37)(27,33,39,30)(28,32,40,36);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,16,13,18)(12,21,22,17)(19,23,20,24)(25,30,31,33)(26,28,35,40)(27,38,39,29)(32,37,36,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,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, 7641579);