#7777944

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	#44228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4613 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)(11,19)(12,16)(13,14)(15,17)(18,24)(20,23)(21,22);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,19)(13,20,24,22)(14,21,18,23)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(14,18)(20,22)(26,31)(29,38)(32,39)(36,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,16,17,19)(13,20,24,22)(14,23,18,21)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,22)(14,17)(15,24)(16,23)(19,21)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,14,15,18)(10,13,17,24)(11,21,12,23)(16,20,19,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=ghi,e^2=gi,f^2=h,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, 7685140);