#8301997

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	#52391	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8209193);