#8772604

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	#40596	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8679800);