#7715829

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	#35005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4556 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)(11,19,20,16)(13,17,24,21)(14,22,23,18)(25,26,30,33)(27,34,38,31)(28,29,39,32)(35,37,36,40);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,22,21,18)(25,27,30,38)(26,31,33,34)(28,36,39,35)(29,37,32,40);(11,20)(13,24)(16,19)(17,21)(25,28,30,39)(26,32,33,29)(27,35,38,36)(31,37,34,40);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,19,20,16)(13,21,24,17)(14,22,23,18)(25,28)(26,29)(27,36)(30,39)(31,37)(32,33)(34,40)(35,38);(9,13)(10,17)(11,14)(12,21)(15,24)(16,18)(19,22)(20,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,34)(35,39);(9,14)(10,18)(11,13)(12,22)(15,23)(16,17)(19,21)(20,24)(25,29)(26,28)(27,37)(30,32)(31,36)(33,39)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7623025);