#7736395

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	#44226	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=ghi,e^2=gh,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, 7643591);