#8914102

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	#52053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8821298);