#7812096

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	#48825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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)(11,19)(12,23)(13,18)(14,16)(15,22)(17,21)(20,24)(25,26,30,31)(27,32,33,39)(28,36,38,29)(34,35,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,26,30,31)(27,32,33,39)(28,29,38,36)(34,40,37,35);(10,16)(15,19)(17,23)(18,24)(26,31)(27,33)(29,36)(34,37);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,27)(26,32)(28,37)(29,35)(30,33)(31,39)(34,38)(36,40);(1,2,5,4)(3,6,8,7)(9,12,14,21)(10,17,16,23)(11,20,22,13)(15,24,19,18)(25,28)(26,29)(27,34)(30,38)(31,36)(32,40)(33,37)(35,39);(1,4,5,2)(3,7,8,6)(9,13,14,20)(10,18,16,24)(11,21,22,12)(15,23,19,17)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7719292);