#7783252

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	#45770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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)(9,10,15,12)(11,16,19,20)(13,23,22,17)(14,18,21,24)(25,26)(27,34)(28,33)(29,35)(30,36)(31,37)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,18)(23,24)(25,27)(26,32)(28,39)(29,36)(30,33)(31,38)(34,37)(35,40);(10,12)(11,19)(14,21)(17,23)(25,28,31,29)(26,33,37,35)(27,36,38,39)(30,32,40,34);(9,12,15,10)(11,20,19,16)(13,17,22,23)(14,24,21,18)(25,27)(26,34)(28,36)(29,39)(30,33)(31,38)(32,37)(35,40);(1,2,5,4)(3,6,8,7)(9,13,15,22)(10,17,12,23)(11,21,19,14)(16,18,20,24)(25,29,31,28)(26,35,37,33)(27,36,38,39)(30,32,40,34);(1,4,5,2)(3,7,8,6)(9,14,15,21)(10,18,12,24)(11,22,19,13)(16,17,20,23)(25,30)(26,36)(27,33)(28,34)(29,32)(31,40)(35,38)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7690448);