#7883007

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	#39863	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790203);